Enjoy this simple dish in winter or early spring when citrus is still at its peak. Free-range chickens are worth the extra expense for their fresh, pure flavor, but this recipe will work with a commercial chicken, too.
Preheat oven to 450°.
Remove and discard giblets and neck. Rinse chicken with cold water; pat dry. Sprinkle 1 teaspoon oregano, 1/2 teaspoon salt, and pepper in body cavity. Place lemon and orange halves in cavity (this will be a tight fit).
Combine remaining oregano and butter. Starting at neck cavity, loosen skin from breast and drumsticks by inserting fingers, gently pushing between skin and meat. Rub butter mixture under loosened skin. Lift wing tips up and over back; tuck under chicken. Tie legs together with cord. Place chicken, breast side up, on a rack coated with cooking spray. Place rack in a broiler pan; add 1 1/2 cups water to pan. Bake at 450° for 50 minutes or until a thermometer registers 170°. Let stand 20 minutes. Remove fruit from cavity and set aside. Remove skin; discard. Sprinkle chicken evenly with 1/2 teaspoon salt.
Place a zip-top plastic bag inside a 2-cup glass measure. Pour drippings from pan into bag; add 1/2 cup water. Let stand 2 minutes (fat will rise to the top). Seal bag; carefully snip off 1 bottom corner of bag. Drain drippings into measuring cup, stopping before fat layer reaches opening; discard remaining fat. Return drippings to broiler pan, and cook over medium heat, scraping pan to loosen browned bits. Simmer 8 minutes or until mixture measures 1/4 cup. Slice chicken; squeeze juice from reserved fruit over chicken. Serve with sauce.