Preheat your oven to 350°F and line a 9x13 inch pan with parchment; lightly grease the sides.
In a medium bowl, whisk together the flour, baking powder, and salt until no streaks remain.
In a large bowl, beat the eggs and granulated sugar with a mixer on high speed for 7 to 9 minutes.
The mixture should become very pale, thick, and fluffy, leaving ribbons on the surface when the beaters are lifted.
Sift the flour mixture over the egg mixture in two additions, gently folding with a spatula after each addition.
Combine the warm milk and vanilla, then gently fold into the batter until just blended and no streaks remain.
Pour the batter into the pan, smooth the top, and tap once to release large air bubbles.
Bake 12 to 15 minutes, until the top is golden and the center springs back when lightly pressed.
Cool completely in the pan, then lift out using the parchment and cut the cake into 1-inch cubes.