Preheat the oven to 350°F. Butter and line two 9 inch round cake pans with parchment paper, lightly butter the parchment paper, then lightly dust the pans with flour.
Into a small bowl, lightly beat the eggs, ¼ cup of milk, and vanilla.
4 large eggs, room temperature, 1 cup whole milk, divided, 2 teaspoons vanilla
Sift the flour, baking powder, and salt into a large mixing bowl or the bowl of your stand mixer. Add the sugar, and mix just until combined.
3 cups (375 grams) cake flour, plus more for dusting the pans, 1½ cups (300 grams) granulated sugar, ½ teaspoon salt, 1 tablespoon baking powder
Add the butter and remaining ¾ cup of milk to the mixing bowl, and beat on low just enough so the flour doesn't spray everywhere. Then increase the speed to medium and beat for 2 to 3 minutes. Scrape the sides of the bowl a few times during this process.
12 tablespoons (1 ½ sticks or 6 ounces) unsalted butter, 1 cup whole milk, divided
Add the mixture of eggs, milk, and vanilla in two or three batches, and beat for about 30 seconds after each addition. Scrape down the bowl each time.
Divide the batter evenly into the two pans and bang them gently on the countertop to spread out the batter and remove any bubbles.
Bake for about 30 minutes, rotating the pans in the oven at about 15 minutes, if needed, to ensure even baking. Start checking for doneness at about 25 minutes. The cakes will have colored slightly and a toothpick inserted into the center will come out clean.
Let the cake layers cool for 10 minutes and then unmold them carefully onto cooling racks for soaking and cooling.
Clean the mixing bowl and beaters now to have them ready when you make the filling and frosting.