Special equipment: a deep fryer or a Dutch oven
For the balsamic glaze: Put the vinegar in a small nonreactive saucepan over high heat and cook until reduced to 1/2 cup, 12 to 15 minutes. Stir in the honey and season with salt and pepper. Transfer to a small bowl and let cool to room temperature.
For the roasted red pepper tomato sauce: Heat the oil in a large Dutch oven over medium-high heat. Add the onion and cook until soft. Add the garlic and red chile flakes and cook for 1 minute.
Add the tomatoes and their juices, salt and pepper, bring to a boil and cook, stirring occasionally, until the tomatoes begin to soften, 20 minutes. Coarsely mash the tomatoes with a potato masher. Add the pureed piquillo peppers and cook over high heat, stirring occasionally. Add the oregano and half of the basil and cook until the sauce is thickened, about 25 minutes more. Taste for seasoning and stir in the parsley. Reserve the remaining basil for serving.
For the eggplant: Preheat the oven to 400 degrees F. Lightly butter the bottom and sides of a 15-by-10-by-2-inch flameproof baking dish and set aside. Heat the oil in a large Dutch oven or deep fryer to 365 degrees F.
Combine the fontina and shredded mozzarella in a bowl and set aside. Combine the breadcrumbs and panko in a large shallow baking dish and season with salt and pepper. Place the flour in a medium shallow baking dish or large plate and season with salt and pepper. Whisk the egg and 2 tablespoons of water together in another medium shallow baking dish and season with salt and pepper.
Season each eggplant slice on both sides with salt and pepper. Dredge each eggplant slice in the flour, tapping off the excess, then dip it in the egg and dredge it in the breadcrumb mixture. Shake off any excess breading and transfer the eggplant to a baking sheet. Repeat with the remaining eggplant slices. Fry the eggplant slices in batches until tender and golden brown on each side and tender, about 2 minutes per side. Remove to a baking sheet lined with paper towels and season with salt.
Lay 8 slices of the fried eggplant in the prepared baking dish, top each slice with about 1/4 cup of the roasted red pepper tomato sauce, 1/4 cup of the combined cheeses and a tablespoon of the remaining chopped basil. Season with salt and pepper. Take 4 of the topped eggplant slices and place them on the other 4 so you have a total of 4 eggplant stacks. Top each stack with one more eggplant slice, 1/4 cup of the sauce, some of the basil and a few thin slices of the fresh mozzarella. Bake until the cheese melts, 8 to 10 minutes. Place under the broiler and broil until the fresh mozzarella is golden brown, about 30 seconds. Top with the remaining chopped fresh basil and drizzle with the balsamic glaze.