Pat chicken dry and season with salt, pepper, and paprika. Sear skin-side down in olive oil over medium-high heat for 5–6 minutes, flip, and cook 3 more minutes. Remove and set aside.
Preheat oven to 400°F. In the same skillet, cook potatoes cut-side down for 3 minutes. Add garlic and thyme, stirring for 30 seconds.
Add butter, chicken broth, and lemon juice, scraping the skillet. Nestle chicken back in skin-side up and add green beans.
Bake 18–20 minutes until chicken reaches 165°F and potatoes are fork-tender.
Spoon sauce over the dish, garnish with parsley, and serve.