Preheat oven to 350 degrees.
Combine cake mix, melted butter, and beaten egg in a large zippered freezer bag. A sandwich bag is too thin. Freezer bags are sturdy. Close bag completely. Massage bag until all ingredients form a smooth dough (10-15 minutes). If preferred, use a stand mixer. Or, combine ingredients in a medium-sized bowl.
Remove cake batter from bag. It will be firm and easy to manage (texture reminds me of play dough). There may be a few dry crumbs. Simply press into batter until smooth. Set aside.
In a medium-sized bowl, add softened cream cheese, melted butter, eggs, lemon juice, and vanilla extract. Use an electric mixer to stir until smooth. Add powdered sugar 1 cup at a time. Stir until filling is smooth.
Lightly butter a 9x5 baking dish or cake pan. Firmly press cake batter (texture similar to play dough) into bottom. Pour filling evenly over dough. Smooth with the back of a spoon.
Bake 40 to 45 minutes. Oven temperatures vary. Decrease to 325 degrees if baking too fast. Avoid over baking. Gooey center will jiggle when done.
Cake is done when a knife inserted in center comes out with a little filling on the blade and a clean tip. If dough is on the tip, return to oven. Check every 10 minutes until knife inserted in center comes out with a clean tip.
Place cake under broiler for 1-2 minutes to brown.
Cake will firm up and stop jiggling after cooling 15 minutes. Serve warm or hot.