Holiday Hens with Fig Glaze and Cornbread Stuffing

• 1 1/2 cups dried California figs, pitted and roughly chopped
• 1/4 cup honey
• 2 tablespoons lemon juice
• 3 cups warm water
• 1/2 cup balsamic vinegar
• 1 stick unsalted butter, for the glaze
• Kosher salt
• Freshly ground black pepper

Stuffing:

• 2 tablespoons extra-virgin olive oil
• 1 small onion, finely chopped
• 1 teaspoon finely chopped rosemary leaves
• 1/4 pound loose sweet Italian sausage
• 2 tablespoons all-purpose flour
• 4 cornbread muffins, torn into pieces, or 1/2 loaf cornbread, cubed (about 2 cups)
• 1 cup rehydrated dried figs, roughly chopped
• Salt
• Pepper
• 1/2 egg, lightly beaten
• 1/4 cup heavy cream
• 1/2 cup chicken stock, homemade or store-bought
• 4 Cornish game hens, 1 to 1 1/2 pounds each
• 4 tablespoons unsalted butter, room temperature
• Kosher salt
• Freshly ground black pepper