Pat chicken breasts completely dry with paper towels. If breasts are thicker than 1 inch, butterfly or pound to even thickness (approximately 3/4 inch). Season both sides with salt, pepper, and paprika if using.
Place cast iron skillet over medium-high heat and add neutral oil. Heat until oil shimmers and reaches approximately 350-375°F.
Carefully place chicken breasts presentation-side down in the hot skillet. Cook undisturbed for 4-5 minutes until a golden crust forms.
Flip chicken using tongs, reduce heat to medium, and add 2 tablespoons of butter to the pan. Allow butter to melt completely.
Tilt the pan slightly to pool the butter, and using a spoon, continuously baste the top of the chicken with the hot butter for 2-3 minutes.
Add remaining butter, minced garlic, and thyme to the pan. Continue cooking for 2-3 minutes, basting frequently to prevent garlic from burning.
Use an instant-read thermometer to verify chicken has reached 163-165°F at the thickest point. If needed, continue cooking and basting until temperature is reached.
Remove chicken from skillet and rest on a warm plate for 3-5 minutes. Meanwhile, continue cooking garlic butter sauce for 30-60 seconds until fragrant but not browned.
Slice chicken against the grain if desired, drizzle with pan sauce, and garnish with fresh parsley and lemon wedges if using.