Generously season chicken breasts on both sides with Cajun or Creole seasoning, plus salt if seasoning mix is salt-free, and pepper. Mist or spray a large, heavy-bottomed skillet like a cast iron skillet with extra virgin olive oil or nonstick spray over high heat then add chicken breasts and sear until golden brown, 1-2 minute per side. Chicken does not need be cooked through. Remove to a plate then set aside.
Melt butter and olive oil in the skillet then add bell pepper, onions, and mushrooms, season with salt and pepper, and saute until caramelized and tender, 6-7 minutes, stirring often to ensure vegetables don't burn. Add chicken breasts back into skillet then pile vegetables on top. Add chicken broth then boil until liquid is reduced by a little more than half and chicken breasts are cooked through, stirring sauce every so often.
Meanwhile, make the parmesan grits. Bring chicken broth and salt to a boil in a small saucepan then add grits and turn heat down to medium-low. Whisk almost constantly for 3-4 minutes or until grits are thickened but still creamy, lifting the pan for a few moments if the grits start to spatter. Remove pan from heat then stir in butter and parmesan cheese. Place a lid on top then set aside. Taste then add salt if needed before serving.
Scoop grits into bowls then add chicken and vegetables on top. Spoon pan sauce on top then sprinkle with green onions and serve.