Preheat the oven to 350°F (175°C). Grease and flour an 8×8-inch baking dish.
In a medium bowl, whisk together the flour, baking powder, and salt. Set aside.
In a large bowl, cream the softened butter and 1 cup sugar together until light and fluffy, about 3 minutes.
Add the 2 whole eggs one at a time, beating well after each addition.
Mix in the lemon zest, lemon juice, and vanilla extract.
Alternately add the dry ingredients and milk to the butter mixture, beginning and ending with the dry ingredients. Mix until just combined.
Pour the batter into the prepared baking dish and spread evenly.
Bake the cake for 25-30 minutes, or until a toothpick inserted in the center comes out clean.
While the cake is baking, prepare the lemon pudding topping. In a medium saucepan, whisk together the egg yolks, 3/4 cup sugar, lemon juice, cornstarch, and water until smooth.
Cook the mixture over medium heat, stirring constantly, until it thickens and comes to a gentle boil, about 5-7 minutes.
Remove from heat and let cool slightly.
In a separate clean bowl, beat the egg whites until stiff peaks form.
Gently fold the beaten egg whites into the slightly cooled lemon pudding mixture until fully combined.
Once the cake is done baking, remove it from the oven and immediately spread the lemon pudding mixture evenly over the hot cake.
Return the baking dish to the oven and bake for an additional 15 minutes, or until the pudding topping is set and lightly golden.
Remove from the oven and allow the lemon cake lemon pudding to cool completely before serving.