Combine first 6 ingredients in a shallow dish or large heavy-duty zip-top plastic bag, gently squeezing to blend; add cornish hens. Cover or seal, and chill 30 minutes, turning occasionally.
Remove cornish hens 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 cornish hens over empty side; grill, covered with grill lid, 45 minutes or until done.
Pour reserved marinade into a small saucepan. Reserve 1/4 cup chicken broth, and add remaining chicken broth to mariande. 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 cornish hens, garnish, if desired.