Trim any excess fat from the chicken breasts and cut them into 1-inch cubes on a cutting board.
Heat 2 tablespoons olive oil in a large Dutch oven or 5–7 quart skillet over medium-high heat.
Add 1 tablespoon minced garlic and sauté until fragrant, about 1 minute.
Add the cubed chicken to the pan; season with 1 teaspoon of the Italian seasoning, 1/2 teaspoon salt, and 1/4 teaspoon pepper. Cook, stirring occasionally, until the chicken is no longer pink on the outside and begins to brown, about 4–5 minutes.
Pour in the chicken broth, marinara sauce, and 1 teaspoon balsamic vinegar, then add the remaining 1 teaspoon Italian seasoning and stir gently to combine.
Add the 16 ounces penne pasta, pressing it gently so it is submerged in the liquid.
Bring the pot to a boil over medium-high heat, then reduce heat to a simmer. Cover and cook, stirring every 5 minutes, until the pasta is tender, about 13–15 minutes.
When the pasta is tender, sprinkle 1 cup shredded mozzarella and 1/4 cup grated Parmesan evenly over the top. Reduce heat to low, cover, and let sit 2–3 minutes to melt the cheese.
Uncover, gently stir if desired, then serve warm.
