Heat oil in a large skillet over medium-high and cook chicken until golden and cooked through; remove and set aside.
Add onion and peppers to skillet; sauté 4–5 minutes until softened, then stir in garlic and spices for 30 seconds.
Add pasta and chicken broth, scraping any browned bits; bring to a simmer and cook until pasta is nearly al dente, about 8–10 minutes.
Stir in cream and cooked chicken; cook 1–2 minutes more until sauce thickens.
Remove from heat, fold in cheese until melted, and season to taste. Serve with cilantro and lime.