Preheat the oven to 350 degrees. Grease and flour a bundt pan with at least 10 cup capacity.
In a large bowl or the bowl of a stand mixer, combine the sugar, eggs, oil, and vanilla. Beat on medium speed (I use number 4 setting on my stand mixer) for 2 minutes. Add the flour, soda, salt, and cinnamon and stir just until barely combined. Fold in the carrots. Pour the mixture into the bundt pan. Bake for approximately 35-40 minutes or until a toothpick or cake tester inserted comes out clean. Allow to cool for about 10 minutes in the pan and then invert on to a cooling rack. Allow to cool completely if you want the frosting to drizzle evenly.
To prepare the brown butter frosting
Add the cubed butter to a small saucepan over medium heat. Stir occasionally until the butter is melted to ensure that it melts evenly. Once the butter has melted, it will begin to bubble.Continue to stir regularly. Small golden flecks will begin to form on the bottom of the pan. Stir or whisk gently continuously at this point as the butter is beginning to brown. The bubbling should subside at this point and you should notice the butter takes on a golden tint and begins to smell nutty. Do not let it burn, but stir until golden brown and then whisk the mixture into a medium sized bowl to cool slightly.
Once slightly cooled, about 5 minutes, add the sugar and salt, whisking to slightly combine. Add a tablespoon of milk, and continue to add small amounts until the mixture is the right consistency. I like my glaze a bit thicker so that it drizzles nicely on the sides of the cake. To test for this viscosity, drag your whisk through the mixture- the line in the mixture should come back together almost completely in about ten seconds. Pour the glaze over the cake.
I like to use a finely grated carrot so that it evenly disperses in the cake and stays extra moist. Do not use store-bought pre-shredded carrots.
You can also bake this cake in round cake pans (3). Baking time will differ.
If your brown butter frosting sets out for too long it will start to crust over. Heat for 5-10 seconds in the microwave and whisk to prepare it for the cake again.
If you burn your butter, pour out and start again! There are great videos on Youtube on browning butter.
For help on browning butter, see my post here: http://thewoodandspoon.com/you-need-to-know-how-to-brown-butter/