Step 1

Mix rosemary, half of garlic, and 2 Tbsp. oil in a small bowl; rub over flesh side of chicken. Cover; chill 1–12 hours.

Step 2

Place a rack in lower third of oven; preheat to 400°. Pat skin side of chicken dry; season with salt and pepper. Heat remaining 1 Tbsp. oil in a large ovenproof skillet over medium-high heat. Add chicken, skin side down, and cook until golden brown, about 5 minutes.

Step 3

Transfer to oven; roast until an instant-read thermometer inserted in the thickest part of thigh registers 165°, 20–25 minutes. Transfer chicken, skin side up, to a plate.

Step 4

Cook shallot, thyme, and remaining half of garlic in skillet, stirring occasionally, until softened, about 2 minutes. Add broth and cook, scraping up browned bits, until reduced by half, about 3 minutes. Gradually whisk in butter. Serve chicken with pan sauce.