First, brown the butter: take your ½ cup of butter, cut it into slices, and place them into a skillet over medium heat. Melt the butter, stirring frequently. It will begin to sputter and crackle. Continue stirring continuously to prevent burning. After a while, the crackling will subside and butter will begin to foam. Keep stirring. Once you see brown flecks and the melted butter turns a rich amber color, immediately remove it from heat and pour the butter into a separate, heatproof bowl.
Let the butter cool completely until it turns into a solid. You can speed this up by putting the butter in the fridge, but make sure it does not get too cold. You want it to be soft and at room temp--similar to if you were baking with normal butter.
Once the butter is ready, preheat your oven to 350°F. Grease and line three, 6" round cake pans.
Prepare the buttermilk by bringing it to room temperature (10 seconds in the microwave works fine), and then stirring in the vanilla extract. Set this aside for now.
Add the browned butter, granulated sugar, and brown sugar into the bowl of an electric mixer fitted with the paddle attachment. Cream the butter and sugars together for 1-2 minutes on medium-high speed to combine them. Then, add 2 tbsp of dulce de leche and beat for another minute.
Scrape down the sides of your bowl, then on low speed, add the eggs, one at a time. Then, turn up the speed to medium high and beat for 1-2 minutes until the mixture is light and creamy. Scrape down the sides of your bowl again.
Now, time to add in the oil. While running your mixer on medium-low, add the oil, one tablespoon at a time, waiting around 10 seconds between each addition. Then, beat the oil in for 2-3 minutes on medium-high speed until you have a very fluffy mixture that's light in color.
On low speed, slowly stream in the buttermilk, then turn up the speed and beat for another minute until combined.
In a separate bowl, whisk the flour, baking powder, and salt together. Then add these to your mixer and fold on low speed until the flour is almost fully combined. Use a flexible spatula to finish folding the batter by hand.
Transfer the batter to your cake pans (use a measuring cup to divide evenly) and bake for 30-35 minutes, or until the cakes are set. Let the cakes cool for 15 minutes in the pan before transferring them to wire racks to cool completely.
Chill the the cakes in the fridge or freezer for at least 1 hour before assembling to make it easier. You can pause here and keep them in the fridge/freezer overnight as well.