Preheat oven to 350°F. Grease and flour a 9x13-inch pan or line with parchment.
Warm milk with butter in a small saucepan until butter melts. Keep warm but do not boil.
Whisk flour, baking powder, and salt in a bowl until evenly combined.
Beat eggs and sugar on high until thick, pale, and doubled, about 3 to 4 minutes.
Mix in vanilla on low speed just to combine.
Sprinkle dry ingredients over the egg mixture and fold gently until almost smooth.
Stream in the warm milk-butter mixture while mixing on low until the batter is silky.
Pour batter into the pan and tap once to release air bubbles.
Bake until golden and a toothpick comes out clean, 30 to 35 minutes.
Cool 10 minutes in the pan, then turn out to finish cooling. Dust with powdered sugar.
Notes
For a cozy twist, add a pinch of grated nutmeg to the batter, or swap vanilla for almond extract. Serve plain with a dusting of powdered sugar, or top slices with fresh berries and lightly sweetened whipped cream. Store covered at room temperature up to 3 days, or wrap and freeze slices for up to 2 months.This recipe is an original creation inspired by classic Old-Fashioned Hot Milk Cake Recipe flavors. All ingredient ratios and instructions are independently developed.