Position rack in middle of oven; preheat broiler to high.

Place panko in a shallow dish and dredge cutlets in it, pressing gently to fully coat.

Heat 1 ½ tablespoons oil in a large cast-iron skillet over medium-high heat; add half the cutlets and cook until crispy, 3 to 4 minutes per side. Transfer to a plate. Repeat with 1 ½ tablespoons oil and the remaining cutlets. Season with salt and pepper.

Add the remaining 1 tablespoon oil and garlic to the pan; cook, stirring often, until softened and fragrant, about 1 minute. Stir in tomato sauce and sugar and bring to a boil over medium-high. Reduce heat to medium-low and simmer, stirring occasionally, until slightly thickened, about 5 minutes.

Nestle the cutlets into the sauce in the pan. Top with ½ cup basil, mozzarella and Parmesan. Transfer the pan to the oven.

Broil until the cheese is bubbling, 2 to 3 minutes. Let cool for 3 minutes before serving. Top with the remaining basil.