Pat chicken thighs dry with paper towels, trim away any excess skin and fat, and season both sides with salt and pepper, to taste.

Heat olive oil in a large skillet over medium heat. Cook chicken thighs in the hot skillet, skin side down, until browned, 5 to 7 minutes. Remove from the skillet and keep warm.

Peel carrots and cut into 3-inch-long pieces, about 1/2-inch thick. Scrub red potatoes and cut larger potatoes in half. Place carrot pieces, potato pieces, chopped celery, herbes de Provence, and garlic in a slow cooker. Season with salt and pepper to taste, and stir to distribute vegetables and seasonings evenly.

Place browned chicken thighs on top, skin side up. Cover and cook on Low until chicken is no longer pink at the center and juices run clear, and vegetables are tender, 5 to 6 hours. Do not open the lid during the cooking time.