Slice the chicken breasts in the half through the equator to create 4 fillets. Cover with plastic wrap and pound to an even thickness (use a meat mallet or side of a can). Pat dry. Whisk 2 tablespoons flour and all of the chicken seasoning together in a shallow dish. Dredge each cutlet in the mixture, shake off any excess, then transfer to a dry surface.
Melt 2 tablespoons butter in 2 tablespoons oil in a large saucepan (I love my braiser) over medium-high heat. Once hot, add chicken and cook each side for 3-4 minutes (depending on thickness) until golden and cooked to 160 degrees F on an instant read thermometer. Transfer chicken to a plate, don't wipe out the skillet. Melt 1 tablespoon butter in the chicken drippings over medium heat. Whisk in flour and cook for 2 minutes. Reduce heat to low and gradually whisk in the chicken broth followed by the green chilies, jalapenos and all seasonings.
Bring the sauce to a simmer while scraping up any brown bits on the bottom of the pan; simmer until sauce thickens almost to desired consistency (it will continue to thicken once you add the sour cream and cheese), stirring often.
Reduce heat to low and stir in the cheddar cheese until melted, followed by the sour cream. Stir in lime juice and cilantro. Add chicken back to the pan and spoon some sauce over top. Serve with desired toppings such as Cotija, tomatoes, jalapenos, cilantro, avocados, crushed tortilla chips etc.