Preheat oven to 400 degrees. Using kitchen shears, cut along both sides of the backbone to remove; discard or reserve for broth, if desired. Open the chicken's legs and spread the bird down flat, skin side up. Press down firmly on the breastbone to flatten. Pat dry with paper towels; season generously with salt and pepper.
Heat a large ovenproof skillet, preferably cast-iron, over high heat. Add 1 tablespoon olive oil and 1 tablespoon butter. Add chicken, skin-side down, to skillet. Let brown without moving, about 3 minutes. Turn chicken, taking care not to break the skin; transfer skillet to oven.
Roast chicken until golden brown and cooked through or until an instant-read thermometer inserted into the thickest part reaches 165 degrees, about 45 minutes or so. Transfer chicken to a cutting board and let stand 10 minutes.
Add 1 tablespoon lemon juice and remaining tablespoon butter to pan, swirling to combine; set aside. In a small bowl, whisk together remaining 3 tablespoons olive oil, 2 tablespoons lemon juice, red-pepper flakes, garlic, and pinch of salt. Cut chicken into pieces and serve immediately drizzled with olive oil mixture and pan sauce.