Melt the butter in a large saute pan over medium-low heat. Toast the breadcrumbs until golden brown, and then transfer to a large mixing bowl.
Add the olive oil to the same saute pan over medium-low heat and cook the eggplant, tomatoes and garlic until the eggplant is soft and will mash easily, about 15 minutes. Remove from the heat and gently mash with a potato masher until about 3/4 of the way smooth. It is important not to mash all the way, for texture.
Combine the tomato and eggplant with the breadcrumbs. Add in the basil, mozzarella, ricotta, Parmesan, Italian seasoning, egg and some salt and pepper and mix thoroughly.
Roll out your wonton wrappers. Place a dollop of filling into the center of each wonton using a tablespoon measure. Fold over the ends to make an enclosed envelope, and then secure by either scalloping or pressing the prongs of a fork into the edge.
Heat some vegetable oil in a saute pan.
Place the dumplings in a single layer in a large frying pan (you may need to work in batches). Add a few tablespoons water and heat slowly, covered, until the water has evaporated, about 5 minutes. Uncover and add 1 tablespoon of the hot oil to fry the bottoms of the dumplings.