Prepare brine. Put chicken breast-down in brine, weight with a small plate, cover, and chill at least 8 and up to 12 hours.
Drain chicken (discard brine) and pat completely dry inside and out. Rub chicken all over with oil, then set breast-up on a V-shaped rack in a 12- by 17-in. roasting pan; tuck wing tips under if you like. Let chicken stand at room temperature 30 minutes. Meanwhile, preheat oven to 400°.
Roast until an instant-read thermometer inserted horizontally through thickest part of breast reaches 160°, 45 to 55 minutes for a smaller bird or 1 to 1 1/4 hours for a larger one; after 30 minutes, if any areas start to get dark, tent them with foil. Transfer chicken to a warm platter and let stand in a warm place about 10 minutes.
Serve chicken with accompaniments.
*To find small (uncooked) chickens, try markets that roast rotisserie birds (which are small) on-site.
Make ahead: Up to 1 day, brine chicken as directed, then drain and chill.
Nutritional analysis per serving for Mexican Chicken: 411 Cal., 55% (224 cal.) from fat; 43g Protein; 25g Fat (8 g sat.); 9g Carbo (5g Fiber); Sodium n/a; 134mg Chol
Note: Nutritional analysis is per serving for Chinese Chicken.