Dust the feta cheese slices with flour and season both sides with salt and pepper.
One by one, dip the slices in the egg and then coat with bread crumbs. Firmly press the bread crumbs into the feta cheese. Dip the slices into the egg and the bread crumbs for a second time. Chill the breaded slices in the refrigerator for at least 30 minutes.
Heat the oil in a deep frying pan and fry the cheese for 3-4 minutes until golden. Place the fried feta on a paper-towel lined plate to absorb any excess oil.
Heat a cast iron skillet until very hot. Brush the eggplant slices with olive oil, season with salt and pepper and cook until golden (about 3 minutes each side).
Toast the burger buns and toss the mixed salad leaves with the balsamic vinegar.
Smear mayonnaise on one half of the burger bun and top with 2 eggplant slices and sun-dried tomatoes. Add the breaded feta cheese, a few salad leaves, and cover with the other half of the burger bun and serve.