Bring a large pot of salted water to a boil. Cook pasta until al dente according to package instructions. Reserve about 1/2 cup of the pasta cooking liquid, then drain the pasta. Toss the pasta with a bit of olive oil to prevent sticking and set aside.

Meanwhile, heat 2 tablespoons of the olive oil over high in a large, deep skillet. Add the chicken, 1/4 teaspoon salt and 1/4 teaspoon black pepper. Cook until the chicken is no longer pink in the center, about 3 minutes per side. Remove to a plate, then carefully wipe the pan with a paper towel, being careful not to burn your fingers.

Reduce the heat to medium and add the remaining 1 tablespoon olive oil. Add the tomatoes, garlic, and remaining 1/4 teaspoon salt and 1/4 teaspoon black pepper. Let cook 1 minute, then reduce the heat to low and cook for 5 to 7 minutes more, stirring occasionally, until the tomatoes begin to soften and wrinkle. Add the balsamic vinegar and cook 30 seconds.

To the skillet, add the drained pasta, reserved chicken, and spinach and let cook just until the spinach begins to wilt. If the pasta seems too dry, add a bit of the reserved pasta water. Remove from heat and stir in the basil and mozzarella. Serve immediately.