Bring first 4 ingredients to simmer in heavy medium saucepan over medium-high heat, stirring until sugar dissolves. Boil until reduced to 1 cup, stirring frequently, about 20 minutes. Puree sauce in processor. Strain into small bowl. (Sauce can be made 1 day ahead. Cool, cover and chill.)

Preheat oven to 400°F. Place game hens, cut side down, on small baking sheet. Sprinkle with salt and pepper. Bake 30 minutes. Increase oven temperature to 450°F. Continue baking until juices run clear when thickest part of thigh is pierced, about 20 minutes longer. If cooking a whole chicken, cook at 425°F for 15 minutes, then lower the oven temperature to 350°F. Cook until done (ours took about 1 ½ hours).

Meanwhile, bring broth to simmer in heavy small saucepan; reduce heat to low and keep hot. Heat a swirl of olive oil in heavy large saucepan over medium-high heat. Add sweet potato, and cook until browned and tender. Remove from pan and allow to cool, then put in a food processor and puree (if it is too thick to properly puree, add some olive oil and/or broth to thin).

Return pan to burner, and heat another swirl of olive oil if needed. Add onion and ginger and cook until beginning to brown, stirring often, about 5 minutes. Add thyme; stir 1 minute. Add rice and stir 1 minute. Add wine; stir until almost absorbed, about 2 minutes. Reduce heat to medium. Add 1 cup hot broth and cook until almost all liquid is absorbed, stirring often. Continue adding hot broth 1 cup at a time and stirring frequently until rice is almost tender, 15-20 minutes. When almost al dente, add sweet potato puree and stir until combined, adding more liquid if necessary to make it the proper consistency. Season with salt and pepper.

Spoon risotto onto plates. Top with hen, and drizzle with cranberry sauce.