Preheat the oven to 400 degrees. Cut off any fat, bone, and gristle remaining on the chicken thighs. Place two thighs between two sheets of plastic wrap. Pound them lightly with the toothed side of a meat mallet to a more or less even thickness. Pound the remaining two thighs the same way. Season the chicken thighs with salt and pepper. Spread the flour and bread crumbs on two separate plates. Beat the eggs in a wide shallow bowl until blended. Dredge the chicken in flour to coat lightly, and tap off excess flour. Dip the floured thighs in the beaten egg, letting the excess egg drip back into the bowl. Transfer the chicken, one piece at time, to a plate of bread crumbs, and turn it to coat with bread crumbs, patting gently and making sure that each thigh is well coated.

Heat the vegetable oil in a wide heavy skillet over medium-high heat until a corner of one of the coated thighs gives off a lively sizzle when dipped in the oil. Add to the oil as much of the breaded chicken as fits without touching. Fry, turning once, until the chicken is golden on both sides and cooked through, about 8 minutes. Remove it to a baking sheet lined with paper towels, and drain well. Cook the remaining chicken (if necessary), adding oil and keeping it at the correct temperature.

Remove the paper towels from the baking sheet. Top each chicken thigh with overlapping slices of tomato, dividing the tomatoes evenly. If you like this dish more traditional, top the chicken with some tomato sauce as well (see below). Drape the sliced cheese over the tomatoes to cover the chicken completely.

For the sauce: Heat 3 tablespoons of the olive oil in a large skillet over medium heat. When the oil is hot, add the garlic, and cook, shaking the pan, until it is golden brown, about 2 minutes. Carefully slide the chopped tomatoes into the skillet, season with salt and pepper, and cook until the sauce is slightly thickened, about 10 minutes. Remove the pan from the heat, set it aside, and fish out the garlic.

Bake the chicken until the cheese is lightly browned, about 10 minutes. While the chicken is baking, reheat the tomato sauce to simmering, stir in the basil, and taste, seasoning with salt and pepper if necessary. Spoon the sauce onto a heated platter or onto individual plates, and place the baked chicken over the sauce.