Peel eggplant and cut into 8 slices approximately ½ inch thick and season with salt & pepper on both sides.
Heat half the olive oil in a large deep skillet over medium heat until the oil shimmers. Fry the eggplant in 2 batches until both sides are nicely browned and softened but not mushy. Place fried eggplant on a paper towel lined plate to absorb any excess oil.
Heat the remaining olive oil in the skillet.
Place a slice of pepper on 4 of the eggplant slices, sprinkle with feta cheese and cover with the remaining eggplant slices. Dip each eggplant sandwich in the beaten egg and coat with bread crumbs.
Gently fry the eggplant sandwiches in the hot oil; 2 to 3 minutes on each side until browned and transfer to paper towel lined plate. Allow to cool enough to touch (but still hot), carefully wrap in parchment paper and serve.