2 cups kosher or 1 cup table salt
6 Cornish hens (each 1 to 1 1/2 pounds), trimmed of extra fat, giblets removed, rinsed well
3 cups prepared stuffing, heated until very hot (120°F to 130°F / 50°C to 55°C in microwave)
6 tablespoons balsamic vinegar
3 tablespoons olive oil
1/4 cup dry vermouth or white wine
Dissolve salt in 5 quarts of cold water in small clean bucket or large bowl. Add hens breast side down; refrigerate 2 to 3 hours. Remove, rinse thoroughly, pat dry, and prick skin all over breast and legs with point of paring knife.
Adjust oven rack to middle position and heat oven to 400°F (205°C).
Spoon 1/2 cup of hot stuffing into cavity of each hen; tie legs of each hen together with 10-inch piece of kitchen twine. Leaving as much space as possible between each bird, arrange them breast side down and wings facing out, on large (at least 19 by 13 inches) wire rack, set over an equally large roasting or jelly-roll pan. Whisk balsamic vinegar and oil in small bowl; set aside.
Roast until backs are golden brown, about 25 minutes. Remove pan from oven, brush bird backs with vinegar and oil glaze (reblending before each bird), turn hens breast side up and wings facing out, and brush breast and leg with additional glaze. Return pan to oven, add 1 cup of water, and roast until meat thermometer inserted into stuffed cavity registers about 150°F (65°C), about 15 to 20 minutes longer.
Remove pan from oven again, brush birds with remaining glaze, return pan to oven, add another 1/2 cup water to pan and increase oven temperature to 450°F (230°C). Roast until birds are spotty brown and stuffed cavity registers 165°F (75°C), 5 to 10 minutes. Remove birds from oven and let rest for 10 minutes.
Meanwhile, pour hen jus from roasting pan into small saucepan, spoon off excess fat, add vermouth or wine, and simmer over medium-high heat until flavors blend, 2 to 3 minutes. Drizzle about 1/4 cup sauce over each hen and serve, passing remaining sauce separately.
Circulon Nonstick Bakeware 17-Inch by 13-Inch Roaster with U-Rack