Season the chicken. Place the chicken in a bowl and season with thyme, smoked paprika, about 1/2 teaspoon salt, and black pepper. Toss until evenly coated.
Brown the chicken. Melt 1 tablespoon butter in a deep 12-inch cast-iron skillet over medium-high heat. Add the chicken in a single layer and cook undisturbed for 2–3 minutes. Turn and continue cooking for 5–6 minutes total, until lightly browned but not fully cooked. Transfer to a plate.
Cook the bacon and vegetables. Reduce the heat to medium. Add the chopped bacon, onion, carrots, and remaining 3 tablespoons butter. Cook for 5–6 minutes, stirring occasionally, until the onion softens and the bacon begins to render. Add the garlic and cook for 30 seconds.
Make the chowder base. Sprinkle the flour over the bacon and vegetables. Stir continuously for 1 minute. Slowly pour in the chicken broth while stirring and scraping the browned bits from the bottom of the skillet.
Add the potatoes and chicken. Add the diced potatoes and return the browned chicken to the skillet. Bring to a gentle simmer, then reduce the heat to medium-low.
Simmer until tender. Cover the skillet with a lid or foil and cook for 15–18 minutes, stirring once or twice, until the potatoes are fork-tender and the chicken is cooked through.
Add the cream and cheddar. Turn the heat to low, or remove the skillet from the burner for 1 minute. Stir in the heavy cream. Add the cheddar one handful at a time, stirring until melted before adding more. Do not allow the chowder to boil.
Finish and serve. Taste and add the remaining salt if needed. Let the chowder rest for 3–5 minutes to thicken, then finish with chopped parsley and extra cheddar if desired.