Preheat oven to 350 degrees F. Spray a 9 x 13″ baking dish with cooking spray or grease with butter, and set aside.
Cream together butter and sugar in an electric mixer on medium speed for about 3 minutes.
Add milk, flour, baking powder, salt, and vanilla. Mix on low speed until combined, then medium-high for 2 minutes.
Add egg whites, and mix on medium-high for 2 minutes longer. Spread batter into prepared dish, and bake in preheated oven for 35-40 minutes, or until toothpick inserted into the center comes out clean.
Allow cake to cool completely, then use the end of a chopstick or wooden spoon to poke holes almost to the bottom of the cake all over the surface, about 1″ apart. Use a paper towel to wipe the cake off your spoon after every few pokes.
In a medium bowl, whisk together lime juice and condensed milk. Set aside.
Mix whipping cream on medium-high speed until it starts to thicken. Add in 2 tbsp. sugar, and continue mixing until stiff peaks form.
Gently fold in 1 cup of your glorious whipped cream into the lime mixture, then pour the lime filling over the top of your cake. Spread it all over the top so it fills the holes, covers the top, and oozes over the sides of the cake a bit.
Gently spread the remaining whipped cream over the top of the lime filling. Cover your lime poke cake with plastic wrap, and refrigerate for at least an hour before serving.