Melt butter over medium heat in a small skillet or heavy-bottomed pot.
Continue to heat the butter, stirring periodically, until it turns brown (10-15 minutes). The butter will first boil with big bubbles (it’s not done then). Then, the bubbles will get smaller.
Turn up the heat slightly and monitor closely until the butter looks more brown (that’s when it’s done).
Remove butter from heat and cool to room temperature. Transfer to a shock-proof container and chill until softened, one hour in the refrigerator or 10-15 minutes in the freezer.
Preheat oven to 350 F (175 C).
In a medium-sized mixing bowl or bowl of a stand mixer, mix together sugar, cake flour, baking powder, baking soda, and salt.
Add browned butter and mix on medium-low speed for three minutes.
In a small mixing bowl, whisk together eggs, sour cream, and vanilla extract until smooth.
Add the egg mixture to the flour mixture and beat on medium speed until just combined. Be sure to scrape down the side of the bowl as needed to make sure that everything is incorporated into the batter.
Slowly add milk and mix on low speed until just combined.
Spread batter evenly in a well-greased 9" square cake pan.
Bake for 20 minutes and then test to see if the cake is done. It is done when a toothpick comes out without wet batter stuck to it. If it is not done, test again in four minutes. If it is still not done, test again in another two minutes.
When the cake is done, place it on a cooling rack (or just on your counter if you don't own a cooling rack) to cool to room temperature.
Spread dulce de leche evenly over cooled cake just before serving.
Notes
When you make your own dulce de leche, it can be really thin like a syrup or more of a spreadable consistency like peanut butter. For this cake, it is important that you use a thicker dulce de leche. I use the one that I make in my Instant Pot and it works perfectly.Most of the store-bought dulce de leche that you find will be thick and work just fine.After the dulce de leche sits on the cake for a few hours, it will begin to absorb into the cake. This makes the cake even more delicious, but it doesn't look as pretty because the dulce de leche is no longer sitting on top of the cake.