Combine first 6 ingredients in a shallow dish; add quail. Cover and chill 30
minutes, turning occasionally.
Remove quail from marinade, reserving marinade.
Prepare fire by piling charcoal or lava rocks on one 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 (cooking time for hens will be about 45
to 50 minutes).
Pour reserved marinade into a 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, for 5 minutes.
Whisk together cornstarch and reserved chicken broth until smooth. Whisk into
marinade mixture; boil, whisking constantly, for one minute.