Preheat oven to 425 degrees F, set rack in lower third of oven, and lightly oil the bottom of a roasting pan. Place chicken at center of pan, pat throughly dry with paper towels, then brush or drizzle and rub with 2 tablespoons olive oil. Sprinkle the outside of the chicken evenly with a teaspoon of salt and ½ teaspoon of pepper. Sprinkle the inside with ½ teaspoon each salt and pepper. Stuff the cavity with the quartered lemon, one sprig of rosemary, and 2 unpeeled cloves of garlic. Tie legs together with kitchen twine and tuck wings underneath the body of the chicken.
Scatter vegetables around the chicken in the pan (plus the reserved sprig of rosemary torn in half and the other two cloves of garlic), drizzle with a tablespoon of olive, season with ½ teaspoon each salt and pepper and the pinch of crushed red pepper. Stir to coat evenly. Turn potatoes so the cut sides face down.
Roast for 45 minutes. Remove roasting pan from oven, turn vegetables over, and drizzle the chicken with a little additional oil if dry. If the chicken is already browning well, tent with foil. Return to oven for another half hour or more as needed until chicken is golden brown and an instant read thermometer stuck into the fleshy part of the thigh registers between 175 and 180 degrees F. The juices should also run clear where the thigh has been pierced. If not yet done, continue cooking another 10 minutes and test again.
Remove from oven and using a thin-bladed spatula, scoop vegetables into a serving bowl and cover to keep warm. Let the chicken rest, tented with foil for 15 minutes on a cutting board before carving. Taste the vegetables and adjust seasoning as necessary. Serve hot, warm, or cold and drizzle with pan juices if you like. If you want a pan sauce, see NOTES. Store leftovers for 3-4 days tightly wrapped in the refrigerator. You might also remove meat from bones, use as needed, and freeze the carcass to make chicken stock another day.