In a large nonstick frying pan, heat 1 tablespoon each of the oil and the butter over moderate heat. Season the chicken with 1/4 teaspoons each of the salt and pepper. Cook the breasts until browned and just done, 4-5 minutes per side. Remove the chicken from the pan and let it rest for 5 minutes. Cut into small pieces.

In the same pan, heat the remaining 1 tablespoon oil over moderately low heat. Add the red onion and cook, stirirng occasionaly, until starting to soften, about 3 minutes. Stir in the garlic, Brussels sprouts, broth, and red-pepper flakes. Bring to a simmer and cook, covered, until the Brussels sprouts are just done, about 5 minutes. Add the chicken, lemon juice, parsley, Parmesan, and the remaining 2 tablespoons butter, 3/4 teaspoons salt and 1/4 teaspoons pepper. Remove from heat.

In a large pot of boiling salted water, cook pasta, until just done, about 10 minutes. Drain and toss with the sauce.