Heat 1 tablespoon olive oil in a large nonstick pan over high heat. Add the sliced red onion and bell peppers, season with a pinch of salt and pepper, and sauté 6–8 minutes until bright and crisp-tender.
Add 1 tablespoon minced garlic to the pan with the vegetables and cook 30 seconds, then transfer the vegetables to a plate and set aside.
Prepare and cook the chicken cutlets according to the linked recipe, cooking in batches in the same skillet so they do not overlap. Transfer cooked chicken to a wire rack to keep crispy.
In the same skillet, melt 2 tablespoons butter over medium heat. Add 3/4 cup chicken broth, 3/4 cup heavy cream, 3/4 cup freshly grated Parmesan, 1 tablespoon lemon juice (from the divided lemon), and 1 tablespoon chopped parsley if using. Bring to a gentle simmer and cook 3–5 minutes, stirring often, until the sauce is thick and smooth and the Parmesan is fully melted.
Meanwhile, bring a large pot of water to a boil (about 12 cups). Add 1 tablespoon salt and the 8 ounces angel-hair pasta. Cook 4–5 minutes or until just al dente according to package directions. Reserve 1 cup pasta cooking water, then drain the pasta.
Add the drained pasta and the sautéed vegetables to the skillet with the sauce and toss to coat. If the sauce is too thick, thin it with reserved pasta water starting with 1/4 cup. Taste and adjust with more lemon juice or salt as needed.
Slice or dice the cooked chicken cutlets and arrange on top of the pasta. Finish with extra grated Parmesan and lemon wedges to serve.
