Preheat oven to 400 degrees.
Place each chicken breast between two sheets of wax paper and pound flat with a rolling pin or meat mallet.
Sprinkle the flattened breasts with 1 teaspoon of salt and l/2 teaspoon of pepper.
In a 12-inch, sauté pan, heat the butter and 1 tablespoon olive oil.
Sauté chicken until it is lightly browned on both sides (3 minutes per side).
Transfer chicken to a 9-by-13-inch oven-proof baking dish.
Deglaze the sauté pan with the port, if desired, and when the mixture is reduced, pour it over the chicken.
Refrigerate chicken until ready to bake.
Heat 1 tablespoon of olive oil in the same sauté pan.
Add the eggplant, a few slices at a time, and season it with the remaining salt and pepper.
Add more oil if needed.
Sauté the eggplant until softened (about 2 minutes on each side).
Top the chicken with the cooked eggplant
(At this point the dish may be covered with plastic wrap and refrigerated for up to 2 days.)
Sprinkle the Gorgonzola over the eggplant and bake the pie for 15 minutes, until cheese is melted.
Serve immediately.