Make the sauce: Heat 1 tablespoon olive oil in a medium saucepan over medium heat until shimmering.
Add minced garlic, 3/4 teaspoon kosher salt, 1/2 teaspoon dried oregano, and a pinch of red pepper flakes; cook until fragrant, about 30 seconds.
Stir in the crushed tomatoes and 1/4 teaspoon sugar, bring to a simmer over medium-high heat, then reduce to low and simmer until slightly thickened, about 20 minutes.
Remove sauce from heat and stir in the remaining 1 tablespoon olive oil and 2 tablespoons chopped basil; season to taste with salt and pepper and keep warm.
Prepare the chicken: Sprinkle each chicken piece lightly with about 1/8 teaspoon salt on each side and let rest at room temperature for 20 minutes.
Combine the shredded mozzarella and fontina in a small bowl and set aside.
Set the oven rack about 4 inches from the broiler and preheat the broiler.
In one shallow dish whisk the egg and flour until smooth. In another shallow dish combine grated Parmesan, Panko, garlic powder, 1/2 teaspoon dried oregano, and 1/4 teaspoon pepper.
Pat each chicken cutlet dry. Dip one cutlet into the egg mixture, then press into the Panko mixture to coat evenly.
Heat vegetable oil in a large nonstick skillet over medium-high until shimmering. Fry 3 cutlets at a time until the bottoms are deep golden brown, about 1½–2 minutes, then flip and cook another 1½–2 minutes. Transfer to a paper towel-lined plate and repeat with remaining cutlets.
Place fried cutlets on a rimmed baking sheet, top each with an even portion of the mozzarella-fontina mixture, and broil until the cheese is melted and beginning to brown, watching closely.
Cook the pasta according to package directions; drain and toss immediately with 1 tablespoon butter.
To serve: divide pasta among plates, place a chicken cutlet on top, spoon about 2 tablespoons sauce over each piece, and sprinkle with torn basil.
