Bring a large pot of salted water to a boil. Cook the pasta until al dente according to the package directions. Reserve 1/3 cup pasta water, then drain and set aside.
Heat olive oil in a large skillet over medium-high heat.
Season the chicken with Italian seasoning, 1/2 teaspoon salt, and black pepper.
Add the chicken to the skillet in a single layer. Cook for 6 to 8 minutes, stirring occasionally, until golden and cooked through.
Add the minced garlic and cook for 30 seconds, stirring constantly, until fragrant.
Add the halved cherry tomatoes and remaining 1/2 teaspoon salt. Cook for 3 to 4 minutes, until the tomatoes soften and release some juices.
Add the cooked pasta and reserved pasta water to the skillet. Toss well to coat everything evenly.
Stir in the Parmesan cheese and chopped basil.
Add the fresh mozzarella and gently stir until it begins to soften and melt slightly.
Remove from heat and drizzle with balsamic glaze.
Serve warm with extra basil, Parmesan, and more balsamic glaze if desired.