Combine smoked paprika, garlic powder, onion powder, salt, oregano, thyme, black pepper, and cayenne in a shallow dish; remove and reserve 1 tablespoon of this seasoning for the sauce, then whisk 2 tablespoons flour into the remaining seasoning.
Slice each chicken breast in half horizontally to make four fillets. Cover with plastic wrap and pound to even thickness, then pat dry.
Dredge each chicken fillet in the seasoned flour mixture, shaking off excess, and set aside on a plate.
Heat 2 tablespoons butter and 2 tablespoons olive oil in a large skillet over medium-high heat. When hot, add chicken and cook 4–5 minutes per side until golden and cooked through. Transfer chicken to a plate; do not wipe out the skillet.
If needed, add oil so there is about 1 tablespoon of fat in the skillet and heat over medium-high. Add chopped onion and cook about 3 minutes, then add halved cherry tomatoes, chopped bell peppers, minced garlic, and the reserved 1 tablespoon seasoning; cook about 2 minutes.
Add about half of the chicken broth to the pan, then whisk the remaining broth with 1 tablespoon cornstarch until smooth and pour it into the skillet along with the heavy cream. Bring to a simmer and cook until the sauce thickens, stirring occasionally.
Reduce heat to low, stir in the grated Parmesan until melted, then add the lime juice. Return the cooked chicken to the skillet and heat 1–2 minutes until warmed through. Taste and adjust salt, pepper, or cayenne as desired.