Bring a large pot of salted water to a boil. Cook the bowtie pasta according to package directions until al dente. Before draining, reserve about 1 cup of the starchy pasta water. Drain and set aside.

While the pasta cooks, pat the chicken pieces dry and season generously with salt and pepper. In a large skillet over medium-high heat, warm the olive oil and 1 tablespoon of the butter. Add the chicken in a single layer and cook until golden brown and cooked through, about 4-5 minutes per side. Transfer the chicken to a plate and set aside.

Reduce the heat to medium. Add the remaining 3 tablespoons of butter to the skillet. Once melted, stir in the minced garlic and red pepper flakes. Sauté for about 1 minute until fragrant, being careful not to let the garlic burn.

Pour in the chicken broth, scraping up any browned bits from the bottom of the pan. Let it simmer for 2-3 minutes. Stir in the heavy cream and let the sauce cook for another 3-5 minutes until it thickens slightly.

Remove the skillet from the heat and stir in the grated Parmesan cheese until melted and smooth. Add the cooked chicken and drained pasta, tossing gently to coat everything evenly. If the sauce seems too thick, add a splash of the reserved pasta water to reach your desired consistency.

Serve the pasta in bowls, garnished with fresh chopped parsley.