Prep the produce: Core and thinly slice the cabbage. Slice the onion and bell pepper. Mince the garlic.
Cut the chicken into small, even pieces for quick cooking.
Season the chicken: Toss the chicken with 1/2 teaspoon salt, 1/2 teaspoon black pepper, smoked paprika, and thyme. Set aside while the pan heats.
Sear the chicken: Heat 1 tablespoon oil in a large skillet over medium-high. Add the chicken in a single layer and cook until browned and cooked through, 5–7 minutes.
Remove to a plate.
Soften the veggies: Add remaining oil and the butter to the skillet. Add onion and bell pepper. Cook 3–4 minutes until softened and a bit golden.
Add cabbage and garlic: Pile in the cabbage with a pinch of salt.
Toss and cook, stirring often, until it softens and reduces in volume, 6–8 minutes. Stir in the garlic and red pepper flakes and cook 30 seconds until fragrant.
Bring it together: Return the chicken and any juices to the skillet. Toss with the veggies and taste.
Add more salt and pepper if needed.
Get it cheesy: Sprinkle cheese evenly over the top. Cover the skillet and reduce heat to low. Let it melt for 2–3 minutes until gooey.
Finish and serve: Squeeze a little lemon over the skillet, if using.
Garnish with parsley or green onions. Serve hot straight from the pan.