Preheat oven to 350 degrees. Lightly butter a 9- by 13-inch baking pan.
In a medium bowl, whisk flour, baking powder and salt.
In a large bowl, using an electric mixer, beat eggs and 3/4 cup sugar on high speed until thick and pale, about 4 minutes. Add vanilla and beat to combine. With mixer on low, gradually add flour mixture and beat to combine. With rubber spatula, fold in melted butter, taking care to deflate mixture as little as possible.
Pour batter into prepared baking pan, and bake till top is golden and a toothpick inserted in center comes out clean, 20 to 30 minutes, rotating pan halfway through baking. (Note: A metal baking pan may bake cake faster than a glass or ceramic; let your eye and a toothpick be your guide.)
In a medium bowl, whisk together sweetened condensed milk, evaporated milk and half-and-half. Poke warm cake all over with a wooden skewer or toothpick, then pour milk mixture over top. Let cake sit at room temperature for 15 minutes, then cover with plastic wrap, and refrigerate 4 hours or overnight.