Pulse garlic, mint, oil, and 1 cup parsley in a food processor until finely chopped; season with 1 tsp. salt and ¼ tsp. pepper.
Season chicken with salt and pepper and spread herb paste over flesh side. Place skin side up on a wire rack set inside a rimmed baking sheet; chill at least 3 hours (the drier the skin, the crisper it’ll get).
Place a rack in lower third of oven; preheat oven to 425°. Pat chicken dry and place, skin side down, in a large ovenproof skillet, preferably cast iron. Set over medium-high heat and cook, undisturbed, until skin is golden brown, about 5 minutes.
Add shallots to skillet and transfer to oven. Roast, turning shallots once, 20–25 minutes. Turn skin side up and roast until skin is very crisp and an instant-read thermometer inserted into the thickest part of thigh registers 160°, 8–10 minutes longer. Serve chicken with shallots, parsley, and any pan juices.
Chicken can be rubbed with herb paste 1 day ahead. Keep chilled.