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

While the pasta cooks, pat the chicken pieces dry with paper towels and season generously with salt and pepper. In a large skillet, heat the olive oil and 1 tablespoon of butter over medium-high heat. Add the chicken in a single layer and sear until golden brown and cooked through, about 3-5 minutes per side. Transfer the chicken to a plate and set aside.

Reduce the heat to medium and add the remaining 3 tablespoons of butter to the skillet. Once melted, stir in the minced garlic and red pepper flakes, cooking for about 1 minute until fragrant but not browned.

Pour in the chicken broth and scrape up any browned bits from the bottom of the pan. Let the broth simmer for 2-3 minutes. Stir in the heavy cream and continue to cook for 3-5 minutes, until the sauce begins to thicken slightly.

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

Divide the pasta among bowls and garnish with fresh chopped parsley. Serve immediately.