Combine garlic, ginger, citrus juice, soy sauce, vinegar, brown sugar, maple syrup, dill, oregano, salt, pepper, and two tablespoons of olive oil in a medium bowl.

Place chicken in a large resealable plastic bag and pour in marinade. Seal bag, pressing out any air. Chill at least 12 hours and up to two days.

Place rack in top third of oven; preheat to 425 degrees Fahrenheit. Remove chicken from bag, place on a large, rimmed baking sheet or nine- x 13-inch roasting pan. Scatter orange and lemon quarters around chicken. Snuggle garlic head between chicken. Drizzle with remaining two tablespoons olive oil.

Pour marinade into a small pot. Bring to a boil over high heat, then reduce to medium and cook until marinade is reduced by about half and slightly thickened.

Roast chicken until deep brown and cooked through, 30 to 40 minutes, basting with reduced sauce a few times while roasting.

Serve chicken on platter with roasted oranges and lemons, roasted garlic, and reduced sauce. Alternatively, squeeze the roasted garlic into the sauce and pour over chicken before serving or serve alongside roasted chicken.