Rinse the chicken and pat dry with paper towels. Place in a roasting pan.
In a small bowl, combine the thyme, red pepper, 3/4 teaspoon of the salt, and 1/4 teaspoon of the black pepper.
Brush the chicken with 1 1/2 tablespoons of the oil and sprinkle with the chopped garlic and spice mixture. Arrange the whole garlic cloves and orange halves around the chicken. Roast until the chicken is cooked through, 35 to 40 minutes.
Meanwhile, place the green beans and mushrooms on a baking sheet. Drizzle with the remaining oil and season with the remaining salt and pepper. Roast until the mushrooms are cooked through and the beans are lightly browned, stirring once, about 15 minutes.
Divide the chicken and garlic among individual plates and squeeze the roasted orange halves over the top. Serve with the green beans and mushrooms.