Preheat the oven to 350ºF and spray a 9x13-inch baking pan with nonstick spray. Set aside.
In a large bowl, whisk the flour, baking powder, and salt together. Set aside.
Add the 5 egg yolks, 1.75 cups of sugar, whole milk, melted butter, and vanilla extract to a medium bowl.
Use an electric hand mixer or stand mixer with the paddle attachment set to medium/high speed and mix until the sugar is dissolved. Use a spatula to scrape down the sides and bottom of the bowl as needed. Set aside.
If using a hand mixer, rinse the egg and sugar mixture off of the beaters.
Add the 5 egg whites to a separate medium-sized bowl. Beat the egg whites on high speed using a hand mixer with clean beaters or stand mixer with the whisk attachment, slowly adding the remaining 1/4 cup of sugar. Mix until stiff peaks form, about 6-10 minutes.
Use a spatula to gently fold the egg white mixture into the egg yolk mixture. Be careful not to deflate the egg whites.
Add the dry ingredients to the egg mixture and whisk until just combined.
Pour the cake batter into the prepared pan. Use a spatula to spread it into an even layer.
Bake for 40-45 minutes or until the top is set and a toothpick inserted into the center comes out clean.
Remove the cake from the oven and let it cool for 15-20 minutes.
While the cake bakes, pour the condensed milk, evaporated milk, and half and half into a bowl. Whisk to combine and set aside.
Once the cake is cool, use a fork (a skewer or straw also work) to poke holes all over the top of the cake.
Pour the tres leches mixture onto the cake. (Optional: Reserve 1/2 cup of this mixture for serving. Transfer to an airtight container and refrigerate until ready to serve.)