Mix together all of the Cajun seasonings in a shallow dish large enough to dredge the chicken. Remove 1 tablespoon seasonings to be used later in the sauce. Whisk 2 tablespoons flour into the seasonings in the dish.
Slice the chicken breasts in the half through the equator to create 4 fillets. Cover with plastic wrap and pound to an even thickness. Pat dry.
Dredge each breast in the Cajun flour mixture, shake off any excess, then transfer to a dry surface.
Melt the 2 tablespoons butter in 2 tablespoons oil in a large skillet over medium-high heat. Once hot, add chicken and cook for 4-5 minutes per side (depending on thickness) until golden and cooked through. Transfer chicken to a plate, don't wipe out skillet.
To the oil left in the skillet (don’t wipe out), add enough oil to equal 1 tablespoon if needed. Heat over medium-high heat. Once hot, add onions, and cook 3 minutes; then add tomatoes, bell peppers, garlic and reserved spice mix and cook 2 minutes.
Add half of the chicken broth to the pan (eyeball it) and whisk the remaining chicken broth with cornstarch until smooth. Add to pan along with heavy cream. Bring the sauce to a simmer and simmer until thickened, stirring occasionally.
Reduce heat to low and stir in Parmesan cheese until melted followed by lime juice. Add chicken back to the skillet and cook a minute or so to heat through. Taste and season with additional salt, pepper and/or cayenne pepper to taste (I like more salt). Garnish with freshly parsley and lime juice if desired.