Cook the pasta: Bring a large pot of salted water to a boil. Cook the pasta until al dente according to package directions. Reserve 1 cup of pasta water, then drain.

Season the chicken: Pat the chicken dry. Season with 1 tsp kosher salt, 1/2 tsp black pepper, and the dried oregano.

Sear the chicken: Heat 2 tbsp olive oil in a large skillet over medium-high heat. Add the chicken in a single layer and cook, undisturbed, until golden on one side, 3–4 minutes. Flip and cook another 2–3 minutes, until just cooked through. Transfer to a plate.

Sauté the aromatics: Reduce heat to medium. Add remaining 1 tbsp olive oil. Stir in garlic and red pepper flakes. Cook 30–45 seconds, just until fragrant.

Build the sauce: Add tomatoes and a big pinch of salt. Cook, stirring, until tomatoes soften and release juices, 3–5 minutes. Pour in chicken broth and balsamic vinegar. Simmer 2–3 minutes to slightly reduce.

Combine: Return chicken and any juices to the pan. Add drained pasta and Parmesan. Toss, adding a splash of reserved pasta water as needed to loosen and gloss the sauce.

Finish fresh: Remove from heat. Gently fold in mozzarella and torn basil. Taste and adjust salt and pepper. The mozzarella should soften but not fully melt.

Serve: Plate with extra Parmesan, a drizzle of good olive oil, and, if you like, a light zigzag of balsamic glaze. Serve immediately.