Bring a large pot of salted water to a boil. Cook pasta until al dente according to package directions, reserving 1/2 cup of the starchy pasta water before draining. Set pasta aside.

While the pasta cooks, heat 1 tablespoon olive oil in a large skillet over medium-high heat. Season chicken with salt and pepper. Add chicken and cook 4–6 minutes until browned and cooked through. Remove chicken to a plate.

In the same skillet, lower heat to medium and add the remaining 1 tablespoon olive oil. Add garlic and sauté 30 seconds until fragrant. Toss in cherry tomatoes and cook 4–6 minutes, stirring occasionally, until tomatoes soften and begin to blister.

Stir in balsamic vinegar and chicken broth (or reserved pasta water). Scrape up any browned bits from the bottom of the pan. Let the mixture simmer 1–2 minutes until slightly reduced.

Return chicken to the skillet along with cooked pasta. Add basil, mozzarella, Parmesan, and red pepper flakes if using. Toss everything together gently until the cheese starts to melt and the pasta is coated. If the mixture seems dry, add a splash more pasta water.

Add salt, pepper, and lemon zest as desired. Garnish with extra basil and Parmesan. Serve hot.