Cook pasta according to package directions until al dente. Reserve ½ cup pasta water, then drain.
Season chicken with salt, pepper, and Southwest seasoning.
Heat olive oil or butter in a large skillet over medium heat.
Cook chicken 5–6 minutes, until golden and cooked through (165°F). Remove and set aside.
In the same skillet, sauté bell peppers 3–4 minutes until slightly tender.
Add garlic and cook 30 seconds until fragrant.
Pour in heavy cream and bring to a gentle simmer over medium-low heat.
Stir in parmesan cheese until melted and smooth.
Add cooked pasta and chicken back to the skillet. Toss to coat evenly.
Add reserved pasta water as needed to loosen the sauce.
Finish with lime juice, taste, and adjust seasoning. Garnish if desired and serve warm.