Preheat oven to 350 degrees. Butter and flour a 9x13 casserole dish or baking pan. This is important so the cake doesn't stick! In a large bowl, mix flour, baking powder, and salt until combined. Set aside.
In the bowl of a stand mixer, add 5 egg yolks and 3/4 cup of sugar. Beat with a beater attachment until the yolks turn a very light yellow, about 4-5 minutes. Add milk and vanilla and mix until combined. Pour yolk mixture over the flour mixture and gently fold together just until combined.
Clean the bowl of the stand mixer. Add egg whites and beat on high until soft peaks form. Add 1/4 cup of sugar and continue beating until stiff peaks form. Add beaten egg whites to the rest of the mixture and slowly and gently fold together until it's almost completely combined.
When the cake is done, remove from oven and let it cool in the pan on top of a wire rack until completely cool (about 1 hour). Once cool, flip cake upside down onto a serving platter.
Using a fork, poke hole all over the top of the cake. This will help the leches soak through the cake. Make leches mixture as directed below.