Bring a large pot of generously salted water to a boil. Cook the linguine according to package directions until al dente. Reserve about 1 cup of pasta water, then drain.
While the pasta cooks, season the chicken pieces with smoked paprika, garlic powder, salt, and black pepper.
Heat the olive oil in a large skillet over medium-high heat. Add the chicken in a single layer and cook for 5–7 minutes, stirring occasionally, until golden and cooked through. Transfer the chicken to a plate.
Reduce the heat to medium and add the butter to the same skillet. Stir in the minced garlic and cook for about 30 seconds, just until fragrant.
Add the chopped chipotle peppers and adobo sauce. Stir for another 30 seconds to release their smoky flavor.
Pour in the heavy cream and chicken broth. Stir well, scraping up any flavorful browned bits from the bottom of the skillet.
Reduce the heat to low and gradually stir in the Parmesan cheese until melted and the sauce becomes smooth and creamy.
Return the cooked chicken to the skillet, then add the drained linguine and lime juice. Toss everything together until evenly coated. If the sauce is too thick, add reserved pasta water a little at a time until it reaches your preferred consistency.
Taste and adjust the seasoning or spice level as needed. Garnish with fresh cilantro and additional Parmesan, if desired, and serve immediately.