Slice each chicken breast through the middle to make four thin fillets; place under plastic wrap and pound to even thickness. Pat dry with paper towels.
Drizzle the chicken with a little olive oil and rub each piece with 1 tablespoon jerk seasoning. Set aside to rest while you prepare other ingredients.
Bring a large pot of heavily salted water to a boil and cook the pasta until just al dente. Reserve 1/2 cup pasta cooking water, drain, rinse briefly, and toss with a drizzle of olive oil; set aside.
In a large heavy-bottomed skillet or braiser, melt 2 tablespoons butter with 2 tablespoons olive oil over medium-high heat. When hot, add the chicken and cook 4–5 minutes per side until golden and an instant-read thermometer reads 160°F. Transfer chicken to a cutting board and loosely tent with foil; do not wipe out the skillet. Let rest at least 5 minutes before slicing.
In the same skillet with the pan drippings, add the sliced onion, sliced bell peppers and 1 teaspoon jerk seasoning. Cook over medium-high heat about 2 minutes, then add the minced garlic and sauté 30 seconds until fragrant.
Sprinkle the flour over the vegetables and cook 1 minute, stirring. Reduce heat to low and slowly whisk in the chicken broth, then add the half and half (or chosen substitute). Stir in the chicken bouillon dry. Increase heat to medium-high and simmer, stirring, until the sauce thickens.
Reduce heat to low and, in batches, stir in the shredded cheddar until melted, then stir in the grated Parmesan until melted. Cook 1–2 minutes on low if needed to finish melting. Stir in 1 tablespoon lime juice.
Slice or chop the rested chicken and add it to the sauce, then fold in the cooked pasta. Add reserved pasta water a splash at a time if needed to loosen the sauce. Season with salt, black pepper and cayenne to taste (example: 1/2 tsp black pepper, 3/4 tsp salt, 1/4 tsp cayenne as a starting point).
Serve garnished with additional freshly grated Parmesan, chopped fresh parsley and an extra squeeze of lime juice if desired.