Cook the pasta: Bring a large pot of salted water to a boil. Cook pasta until just shy of al dente.
Reserve 1 cup pasta water, then drain and set aside.
Prep the chicken: Pat chicken dry and season lightly with salt and pepper. Slice thinly against the grain for quick cooking.
Stir-fry the chicken: Heat 1 tablespoon oil in a large skillet or wok over medium-high heat. Add chicken in a single layer and cook 3–5 minutes until browned and just cooked through.
Remove to a plate.
Sauté veggies: Add remaining oil to the pan. Stir in onion, bell pepper, broccoli, and carrot. Cook 3–4 minutes until crisp-tender.
Add snap peas for the last minute. Push veggies to the edges.
Add aromatics: Drop butter in the center. When melted, add garlic and ginger.
Stir 30 seconds until fragrant, then toss with the veggies.
Build the sauce: Pour in cream, soy sauce, oyster sauce, sesame oil, and rice vinegar. Stir and simmer 1–2 minutes until slightly thickened. If you like heat or a touch of sweetness, add chili-garlic sauce and honey now.
Melt in Parmesan: Lower heat to medium.
Sprinkle in Parmesan, stirring until smooth. If the sauce seems too thick, splash in reserved pasta water to loosen.
Combine: Return chicken to the pan with any juices. Add the cooked pasta and toss to coat.
Warm through for 1–2 minutes. If needed, adjust consistency with more pasta water.
Taste and adjust: Season with pepper and a pinch of salt if necessary. The soy and Parmesan are salty, so taste before salting.
Finish and serve: Stir in spinach to wilt, if using.
Top with scallions, sesame seeds, extra Parmesan, and a squeeze of lime for brightness. Serve hot.