Cornish hens aren't game birds, but small, young chickens — the French call them poussins — that are extremely tender and flavorful. You can also make this recipe with one 4-5 pound chicken, just cooking it at 450 degrees for 15 minutes, then turning the temperature down to 375 degrees and cooking for an additional 30 minutes, or until the chicken is done.
For the Brine:
Combine all ingredients in a large pot and bring to a simmer over medium heat. Stir to dissolve the sugar, then remove from heat and let cool. Refrigerate (do this a day in advance).
Trim the birds by removing the wingtips with kitchen shears. Cut away excess skin around the neck cavity and slice off the neck, if still intact. Add the birds to the refrigerated brine, cover, and refrigerate for at least 3 hours or overnight (if you need to weigh down the birds, place a plate on top, and a heavy bowl or can to weigh it down, so they stay submerged).
For the Panettone Stuffing:
Preheat the oven to 350 degrees. Spread the panettone cubes out on two rimmed baking sheets. In a small saucepan, melt 2 tablespoons of butter over medium heat and cook until light brown, about 6 minutes. Remove from heat and add the sage. Season with salt and pepper. Pour the sage butter over the bread and toss gently. Toast the cubes in the oven, turning about halfway through, until light brown, about 15 minutes. Remove from oven and pour croutons into a large bowl.
Meanwhile, place the dried fruit into a bowl and cover with boiling water. Let soak for at least 10 minutes, or until the fruit is soft and plump. Drain and set aside.
Increase the oven temperature to 375 degrees. Heat a large sauté pan or skillet over medium-low heat, add the remaining 2 tablespoons of butter, and cook until it foams. Add the onion, celery, and carrot and sauté until soft, 12–15 minutes. Stir in the plumped fruit and remaining sage, then add the mixture to the croutons and toss. Add 1 1/2 cups of stock to moisten, more if you like softer stuffing, and the beaten egg if you like a firmer stuffing. Season to taste, and set aside.
For the Roasted Lemon-Rosemary Jus:
Preheat the broiler. Cut a small slice off both ends of each lemon, then cut in half, cross-wise. Arrange the lemons, flesh-side up, in a flameproof baking or gratin dish, brush with olive oil, and season with salt and pepper. Broil 6 inches or more from the heat source until browned and soft, then remove and let cool. Squeeze the lemon halves over a fine-mesh sieve suspended over a bowl, pressing the pulp through but discarding the seeds and rinds.
Heat a large sauté pan or skillet over medium-high heat, add 1/4 cup olive oil, and sauté the garlic until light brown. Immediately add the roasted lemon juice, 1 cup stock, rosemary, and parsley. Stir to scrape up browned bits, and then season to taste (if it is too lemony, add the butter and stir).
To assemble and roast the birds:
Preheat the oven to 375 degrees. Remove the birds from the brine and dry the skin well. Using one cup of stuffing per bird, gently stuff the cavities. With kitchen twine, tie the ends of the legs together, then bring the twine around to secure the neck cavity. Rub each bird with olive oil, then season with salt and pepper.
Heat a large roasting pan over high heat and add the 2 tablespoons of oil and the butter. When the butter foams, sear the birds, breast-side down, on the stove top until golden, about 3 minutes. Flip and sear the other side for 3 minutes. Transfer to an oven and roast for 20–25 minutes, or until a thermometer inserted in the thigh reads 160 degrees. Remove from the oven and let them rest 10 minutes. Remove the twine and serve a bird per person.