Preheat the oven to 400F. Put some parchment paper on a large baking sheet (or 2 medium ones for both puff pastry sheets) and spray with non-stick spray. Thaw the puff pastry and lay it out on the parchment paper.
Prepare the eggplant: Slice the eggplant into thin 1/4 inch circles. Set up two small bowls near the stove - one with Italian breadcrumbs, and one with beaten eggs. Add a generous pinch of salt and pepper to the beaten eggs.
Dip the eggplant in egg and then into the breadcrumbs, make sure both sides are evenly coated.
Heat the vegetable oil in a skillet pan and fry the breaded eggplant slices for about 2 minutes on one side, or until nicely browned. Carefully turn over the eggplant slices, add more oil if needed. Fry on the other side for 2 minutes until nicely browned. Remove the eggplant and place on a paper towel to absorb the extra oil. Repeat until all the eggplant is cooked.
Did the puff pastry thaw by now? OK, good. Arrange the eggplant on the puff pastry, leaving about a 1/2 inch edge of puff pastry all around. Add the tomato slices on top. Season with salt and pepper to taste, and top with basil leaves.
Bake at 400F for 10 minutes, then carefully remove the tray from the oven.
Add the mozzarella slices on top, and place in the oven for another 5 minutes, or until the cheese has melted and the puff pastry has puffed up and turned golden brown on the sides. Slice each puff pastry into 4 pieces and top with more fresh basil.
If you have extra fried eggplant, tomato, and mozzarella cheese, you can make a few eggplant-tomato stacks and bake them for the same amount of time as the puff pastries - top with mozzarella cheese for the last 5 minutes.