To make the cake:

1. Preheat the oven to 350 degrees. Lightly grease three 8” round cake pans and line the bottoms with a round of parchment paper.
2. In a large bowl or the bowl of a stand mixer, cream the butter and sugar on medium speed for 5 minutes. Scrape the sides of the bowl and add the egg and egg yolks one at a time, mixing until combined after each addition. Add the vanilla. Scrape the sides of the bowl and stir in half of the dry ingredients on low speed. Add half of the milk and scrape the bowl again. Repeat this process with the remaining dry ingredients and milk. Fold in any unincorporated bits and divide the batter between the three pans. Bake in the preheated oven for about 20-22 minutes OR until a cake tester inserted into the center comes out clean. Allow to cool completely.

To prepare the custard:

1. Pour the half and half into a heavy-bottomed saucepan over low heat and add the vanilla bean, scraping the insides into the liquid. Bring to a simmer and then turn the heat off and allow the vanilla to infuse into the liquid for 15 minutes. Be sure to stir occasionally.
2. Meanwhile, vigorously whisk or beat the yolks with the sugar on medium speed until it lightens in color and becomes slightly fluffy. Add the cornstarch and stir to combine. 
3. After 15 minutes of infusing the half and half, remove the vanilla bean and carefully pour about ¾ cup of the warm liquid into the egg mixture, whisking or mixing quickly all the while to prevent the eggs from curdling. Pour the egg mixture plus the ¾ cup of added liquid back into the saucepan with the remaining half and half. Whisk to combine and then turn the heat to medium-low. Keep stirring until the mixture comes to a low bubble and begins to thicken. Once thickened to a runny mayonnaise consistency, quickly remove from heat. Feel free to strain the mixture with a fine wire strainer as needed. Set aside in a heat-safe bowl and cover with a piece of plastic wrap to cool completely. The custard can be made a day or two in advance. 

To prepare the burnt sugar crunch:

1. Liberally butter a half sheet pan with rimmed sides that is lined with a full sheet of foil.
2. In a medium, heavy-bottomed saucepan combine the sugar and water over medium heat, stirring together occasionally until the sugar dissolves. Once the mixture gets hot enough it will begin to bubble. Increase the heat to medium-high and avoid stirring it any more. You can gently swirl the pan occasionally to keep the mixture from burning in one spot. Continue cooking over heat until the mixture turns amber colored, about 10 minutes. It may barely begin to smoke. Remove the pot from heat and carefully whisk in the baking soda quickly. Dump the mixture out onto the pan and barely spread it out with a spatula. Don’t overwork it though as this will deflate all the bubbles. Allow the mixture to cool completely prior to breaking and using in the custard. You’ll want to make the crunch within a day of frosting the cake and keep it in a sealed bag as the crunch will absorb moisture from the air and get chewy/sticky over time.

To prepare the buttercream and assemble the cake:

1. Prepare a double boiler. Place a small to medium-sized saucepan with an inch of water on the stove. The pot needs to be slightly smaller than the bowl of your stand mixer to ensure that the bottom of the bowl doesn’t touch the water in the pan. Clean the bowl of your stand mixer well to ensure no trace of fat or grease is in the bowl- this can prevent your egg whites from thickening. I like to squeeze the juice of a half of a lemon into the bowl and use a paper towel to wipe down the insides of the bowl as well as the whisk attachment. Once the bowl is clean, add the room temp egg whites and sugar to the bowl. Turn the burner on to medium heat and station yourself at the stove to stir the eggs regularly. Over the course of about 3-5 minutes, the sugar in the egg whites will dissolves. Once the mixture reaches about 160 degrees farenheit (or when you notice the sugar has dissolved in the eggs- you can carefully feel for this as well!), remove the bowl from the double boiler and place it on the mixer. Using the whisk attachment, whip the egg mixture on medium speed (I use 6 on my Kitchen Aid) for about 15 minutes or until the eggs have developed STIFF peaks. The bowl should be room temperature to the touch and glossy peaks should stand up on the end of the whisk when you scoop them out of the bowl. (Note: sometimes my mixer overheats and the warm eggs have a hard time standing up perfectly straight. Use your best judgment. If the bowl is cool and the eggs are standing up but the mixer is really hot it can make your whites droopy. But don’t underbeat! You frosting will not come together if so!) Scrape the whisk clean and then put the paddle attachment on the machine. Running on medium speed, add the butter 1 tablespoon at a time scraping the sides of the bowl as needed. Once the butter has all been added whip for an additional 3 minutes until the mixture has fluffed up. Here’s where things can get dicey: If your butter is too cold, you can end up with a curdled lumpy looking mixture in your bowl. You’ll know if it’s not right. Remove a cup of the mixture from your bowl and put it into another. Microwave that cup of liquid for about 15 seconds to carefully warm and then add it back to the mixer. Whip again to see if it comes together and repeat if it continues to look the same. If your mixture is runny and loose, it could be your meringue or butter was too warm. Place the entire bowl in the fridge to cool down for about 30 minutes and then rewhip. It should come together but repeat as needed. Once light and fluffy, add the vanilla and salt. Then, break off about half of the burnt sugar crunch and process it in a mini chopper or blender until it’s reduced to a sandy powder- avoid clumps! Add ¼ cup of the sandy powder to the buttercream and whip until combined. Feel free to add a tablespoon of cream as needed if the mixture is too thick. Frosting is best use immediately, so get started assembling! 
2. When ready to assemble the cake, use a serrated knife to level the tops of the cakes. Place a small bit of buttercream on an 8” cake board or plate and then place your first layer of cake on top. Put about half of the buttercream into a piping bag fitted with a large round tip (alternatively you can use a quart-sized Ziploc with the end snipped off) and pipe a dam of frosting about 1/2” tall around the perimeter of the cake. Be sure the ends of the dam meet well so that your filling doesn’t squish out the sides of the cake. Spread half of the custard into the center of the dam and then top it all with a second layer of cake. Repeat this process and then use the remaining frosting to ice the cake. Decorate as preferred and enjoy! Cake is best served the day it is assembled and should be stored in the fridge. Allow the cake to come to rom temp before serving.