Cut tomatoes into 1/2-inch slices. Sprinkle with coarse salt and cracked black pepper.
Place flour, eggs, and crumbs in separate shallow bowls. Heat the oil with a dollop of butter, in a medium skillet on medium heat. Dip the tomato slices in the flour, then the eggs, and then crumbs. In batches, fry half the coated slices at a time, for about 3 to 4 minutes per side, or until golden brown.
Coat the remaining tomatoes in the flour, egg and bread crumb mixture, and cook. Remove the tomatoes to a warm serving tray.
Garnish with a dollop of sour cream, crumbled crisp prosciutto, and a sprinkle of fresh chopped chives.