Combine first 6 ingredients in a shallow dish or large heavy-duty zip-top plastic bag, gently squeezing to blend; add quail. Cover or seal, and chill 30 minutes, turning occasionally.
Remove quail from marinade, reserving marinade.
Prepare fire by piling charcoal or lava rocks on 1 side of grill, leaving the other side empty. Place rack on grill. Arrange quail over empty side; grill, covered with grill lid, 30 minutes or until done.
Pour reserved marinade intoa small saucepan. Reserve 1/4 cup chicken broth, and add remaining chicken broth to marinade. Bring mixture to a boil over medium-high heat; boil, stirring occasionally, 5 minutes.
Whisk together cornstarch and reserved 1/4 cup chicken broth until smooth. Whisk into marinade mixture; boil, whisking constantly, 1 minute. Serve with quail; garnish, if desired.
Asian Grilled Cornish Hens: Substitute 4 (1- to 1 1/2 pound) Cornsh hens for quail. Grill as directed 45 to 50 minutes or until done.