Grill or roast the chicken cutlets over medium heat until cooked through, about 4–6 minutes per side depending on thickness; rest briefly, then slice into strips.
Bring a large pot of water to a rolling boil; add 3 tablespoons olive oil and 1/2 teaspoon kosher salt, then add the bowtie pasta and cook for 9–10 minutes until al dente.
Drain the pasta in a colander and rinse with cold water to stop cooking and cool the noodles.
In a large mixing bowl combine the diced red onion, cherry tomatoes, cooled pasta, and sliced chicken.
In a small saucepan, combine 1/4 cup balsamic vinegar, 1/4 cup olive oil, and salt and pepper to taste; simmer over low–medium heat until the mixture reduces by about half and thickens slightly, about 5–8 minutes.
Pour the warm reduced dressing over the pasta mixture and toss to coat evenly.
Add the cubed mozzarella and chopped fresh basil, toss gently, and adjust seasoning with salt and pepper before serving.