How to Make It
Preheat oven to 425°F. Stir together butter, anchovies, garlic, chopped thyme, lemon zest, 1/2 teaspoon of the salt, and 1/4 teaspoon of the black pepper in a small bowl until well combined. Set aside. Place the chicken, breast side up, on a cutting board. Loosen and lift skin from chicken breast with fingers (do not totally detach skin); spread about 2 tablespoons butter mixture underneath skin. Carefully replace skin. Season outside of chicken with 1/2 teaspoon of the salt and remaining 1/2 teaspoon pepper.
Toss together radishes, thyme sprigs, olive oil, and remaining 1/4 teaspoon salt in a small roasting pan. Spread radishes in a single layer. Place chicken in center over radishes. Bake at 425°F for 35 minutes. Reduce temperature to 325°F; add olives, and bake until a thermometer inserted in thickest portion of breast registers 165°F, 30 to 35 minutes. Remove from oven, and dollop remaining 2 tablespoons butter mixture over chicken. Let stand 10 minutes.
Using a slotted spoon, transfer radishes to a medium bowl, and toss with lemon juice and parsley. Serve radishes and olives with chicken.