Bring a large pot of salted water to a boil and cook the pasta until just shy of al dente according to package directions. Reserve 1 cup of the cooking water, then drain and set the pasta aside.
While the pasta cooks, combine the chicken pieces, 1 tbsp olive oil, smoked paprika, 1 tsp garlic powder, 1 tsp cumin, 1/2 tsp salt and 1/4 tsp black pepper in a bowl. Toss well and let sit 8–10 minutes to absorb the flavors (or refrigerate up to 30 minutes).
Heat a large skillet over medium-high heat and add 1 tbsp olive oil and the butter. Once the butter melts and the pan is hot, add the chicken in a single layer. Cook, stirring occasionally, about 6–8 minutes until pieces are browned and cooked through. Transfer the chicken to a plate and tent with foil to keep warm.
Reduce heat to medium. In the same skillet add the sliced red onion and green pepper. Sauté until softened and slightly charred, about 5 minutes, scraping up any browned bits from the bottom of the pan.
Stir in 1 tsp garlic powder, 1 tsp ground cumin, 1 tsp paprika, 1/2 tsp salt, 1/4 tsp pepper and the tomato paste. Cook while stirring constantly for 1–2 minutes to toast the paste and deepen the flavor.
Pour in the chicken stock and use a wooden spoon to deglaze, loosening any stuck bits. Lower the heat to medium-low and whisk in the softened cream cheese until a smooth sauce forms.
Add the shredded cheddar a handful at a time, stirring until melted and glossy. If the sauce looks too thick, stir in up to 1/2 cup of the reserved pasta water to reach a silky consistency.
Return the drained pasta and the cooked chicken to the skillet. Toss gently to combine and warm through; add more reserved pasta water as needed to loosen the sauce so it coats the pasta evenly.
Finish with the chopped parsley and a squeeze of fresh lime (if using) for brightness. Let the pan rest off the heat for 2 minutes before serving so the sauce settles.