Preheat the oven to 350°F. Grease and flour three 6-inch cake pans and set aside
Stir together the milk, sour cream, and vanilla. Set aside.
In the bowl of a stand mixer fitted with the paddle attachment, place the flour, sugar, baking powder, and salt. Whisk to combine. With the mixer on low speed, add the butter, a few tablespoons at a time, until the butter breaks up into small pieces, no larger than a peanut, and gets coated in flour.
Turn the mixer up to medium-low and stream in the milk mixture. Continue to mix until the batter is evenly moistened. Stop the mixer and thoroughly scrape down the sides and bottom of the bowl. The batter will not be smooth
Working in batches with the mixer on medium-low speed, pour in about ⅓ of the eggs at a time. After each addition of the eggs are absorbed into the batter, stop the mixer and thoroughly scrape down the sides and bottom of the bowl. The batter will look lumpy at first, but should smooth out by the final addition.
Evenly distribute the batter between the prepared pans and bake for 23 to 26 minutes, or until a toothpick inserted into the center of the cake comes out clean. Cool the cakes on a wire rack for 10 to 20 minutes before removing them from their pans.