Place the chicken on a large cutting board, breast-side down. Using kitchen shears, cut along each side of the backbone and remove the bone. (Save the bone for making stock at a later date. Place in a freezer bag and freeze.) Flip the chicken over and press firmly in the center of the breasts to flatten. Season both sides of the chicken with salt and pepper.
Line a rimmed baking sheet with parchment paper or foil. Slice the oranges into ¼-inch slices. Slice two of the lemons into ¼-inch thick slices. Place the orange and lemon slices on the prepared baking sheet to create a “rack” for the chicken to sit on. Top the citrus with the thyme springs. Place the chicken on top of the citrus and herbs, skin side up. Tuck the wings under the breasts. Roast for 45 minutes or until the skin is browned and crispy and an instant-read thermometer inserted into the inner thigh reaches 160 degrees. Remove from the oven, place on a cutting board, and let rest for 5-10 minutes.
Place the parsley, cilantro, basil, garlic, vinegar, capers, 1 teaspoon pepper, and anchovies in a food processor. Pulse several times until the herbs are coarsely chopped. With the machine running, drizzle in enough olive oil to make a thick but pourable sauce. Serve with the chicken.