Using a medium-sized skillet, fill about 1/4 with oil and let heat on medium-high heat.
Meanwhile, set up breading station using a small to medium bowl for the eggs, one plate with Italian bread crumbs and another plate for the Panko crumbs. Salt tomato slices with salt (to taste) and dip into egg mixture then in Italian bread crumbs. Dip the tomato in the egg mixture once more and then into the Panko crumbs. Repeat until all tomato slices are breaded.
Place three to four tomato slices into the oil and fry for 1 to 2 minutes and gently turn the tomatoes over using tongs. Allow to cook for 1 to 2 minutes longer or until golden brown. Drain on paper towels. Serve warm.