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.