In the same pan, heat the a splash of olive oil over moderately low heat. Add the red onion and cook, stirring occasionally, until starting to soften, about 3 minutes. Stir in the garlic, Brussels sprouts, stock, 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, season with salt and pepper to your liking. Remove from the heat.