Make the sauce: Heat the olive oil in a large heavy-bottomed pot over medium-high heat. Add the onion and garlic and saute until the onion is translucent, about 10 minutes. Add the carrot and celery, season with sea salt and pepper and saute until all the vegetables are soft, another 10 minutes. Add the tomatoes and oregano and simmer over low heat, uncovered, until the sauce has thickened, about 1 hour. Season to taste with more sea salt and pepper.
Meanwhile, make the chicken: Preheat the oven to 350 degrees F. Set a wire rack on a baking sheet lined with foil. Sprinkle the chicken on both sides with kosher salt and pepper. Combine the flour and some kosher salt and pepper in a medium baking dish. In a second baking dish, whisk the eggs, milk and minced garlic. In a third dish, combine the panko and garlic salt.
Dredge the chicken breasts through the flour mixture, dusting off any excess, then through the egg mixture, letting the excess drip off, and then through the panko, making sure to coat the chicken on both sides. Set the breaded chicken on the wire rack and refrigerate for 30 minutes to allow the breading to set.
Fill a large deep skillet halfway with vegetable oil and heat over medium heat until a deep-fry thermometer registers 375 degrees F. Using tongs, carefully place the breaded chicken in the hot oil, working in batches to avoid crowding the pan. Fry, turning once, until light golden on both sides, about 4 minutes. Return the chicken to the wire rack.
Dollop 2 tablespoons sauce over each piece of chicken and sprinkle evenly with the parmesan and mozzarella. Transfer to the oven and bake until the chicken is cooked through, about 20 minutes. Serve with the remaining sauce on the side.