Preheat oven to 425 degrees, with racks in upper and lower thirds. Toss garlic, bread, cheese, and tomatoes with 3 tablespoons oil; season with salt and pepper. Arrange in a single layer on a rimmed baking sheet. Rub chicken with remaining 2 tablespoons oil; season with salt and pepper. Place on bread mixture, skin-sides up, along with thyme. Roast in lower third, flipping chicken and bread mixture once, 25 minutes. Meanwhile, remove ribs from kale; tear into 2-inch pieces. Drizzle with oil and season with salt and pepper.

Switch oven to broil; remove thyme from baking sheet. Transfer chicken to a plate. Scatter kale over bread mixture; arrange chicken on top. Broil until chicken skin is golden and crisp, 2 to 4 minutes. Transfer chicken to serving plates; toss bread mixture and serve, with more cheese.