Working with 1 breast at a time, hold a long knife parallel to cutting board and cut 2 medium skinless, boneless chicken breasts (about 1 lb.), patted dry, in half, slicing along a long side to make thin cutlets. Arrange all 4 cutlets in an even layer on cutting board and sprinkle all over with 2 tsp. Diamond Crystal or 1¼ tsp. Morton kosher salt. Transfer to a medium bowl and sprinkle with ¼ cup all-purpose flour; toss to coat.

Heat ¼ cup extra-virgin olive oil in a large high-sided ovenproof skillet, preferably cast iron, over medium. Working in 2 batches and adding more oil between batches if needed, lift cutlets from bowl, shaking off excess flour, and carefully lower into skillet. Cook until deep golden brown, about 3 minutes per side (chicken will finish cooking in oven). Transfer to a large plate or baking sheet.

Reduce heat to medium-low. Add 8 garlic cloves, finely chopped, and ½ tsp. crushed red pepper flakes to skillet and cook, stirring constantly, until fragrant, about 1 minute. Add one 28-oz. can whole peeled tomatoes, crushing with your hands as you go, then add remaining 2¼ tsp. Diamond Crystal or 1¼ tsp. Morton kosher salt and 1½ tsp. sugar. Bring to a simmer and cook, stirring occasionally, until sauce is slightly thickened, about 5 minutes. Add one 17.5-oz. package shelf-stable potato gnocchi and cook, stirring constantly, until gnocchi is barely tender, about 2 minutes. Remove from heat.

Heat broiler. Arrange chicken in a single layer over gnocchi. Top chicken with 2 oz. Parmesan, finely grated, followed by 8 oz. part-skim mozzarella, sliced ¼" thick. Transfer skillet to oven and broil until cheese is melted and browned, 5–8 minutes; watch closely to avoid burning.

Serve chicken parm topped with basil leaves if desired.