Bring a pot of generously salted water to boil.
Season the cubed chicken breast with salt and pepper and toss it with your fingers to coat.
Meanwhile, heat 1 tablespoon oil in large skillet over medium heat. When the oil is shiny and ripples easily in the pan, add the chicken pieces in a single layer. They might stick slightly, that's okay. They'll release when they're ready to turn.
Cook, stirring and turning chicken pieces until just cooked through, about 6-7 minutes. When chicken is done cooking, transfer from the skillet to a paper towel on a plate and set aside.
Add remaining 1 tablespoon oil to the same pan over medium low Add carrots and cook until slightly softened, about 2-3 minutes. Then add the rest of the vegetables, basil, and garlic. Stir to combine.
Drop angel hair pasta into the boiling water, and set a timer according to al dente package directions (usually about 4 minutes).
While pasta is cooking, add chicken broth to vegetables. Stir to scrape the brown bits off the bottom of the pan, then reduce to a simmer and add the parmesan cheese. Stir until melted, then add the chicken back to the skillet.
When pasta is done, use a pasta fork or kitchen tongs to transfer it directly from the boiling water to the skillet. Use tongs to twist and twirl the pasta to evenly distribute the veggies and so that the sauce coats the noodles. Adjust seasoning to taste.
Serve topped with additional parmesan cheese.