Adjust oven racks to accommodate both baking dishes at the same time (one for the hens, one for the stuffing). Preheat oven to 350 degrees.
For the hens:
Truss the hens by tucking the wings against the hens, running a piece of cooking twine from the neck of the bird around the breasts, and tie the drumsticks together.
Place the hens breast side up in a shallow baking dish. Rub each hen with butter, and sprinkle with salt and freshly ground pepper. Bake uncovered for one hour.
For the stuffing:
While the hens are baking, prepare the stuffing. Coat a 9-inch by 9-inch baking dish with butter.
In large skillet over medium-high heat, melt butter until foaming. Add onion and celery and sauté until translucent, about 7 to 8 minutes. Meanwhile, in a large bowl whisk the egg. Stir in broth, 1/2 teaspoon salt, and 1/4 teaspoon pepper.
To the skillet, add parsley, sage, thyme, and marjoram until fragrant, about 30 seconds. Transfer to the bowl with the eggs and mix well. Add bread cubes and toss to combine. Transfer to prepared baking dish.
Cover tightly with foil and bake until mostly heated through, about 25 minutes. Remove foil and bake until crispy edges form, about 15 to 20 minutes longer.
For the apple glaze:
While the hens and stuffing bake, prepare the apple glaze in a small saucepan. Bring the apple cider to boil and cook until reduced by half (1 cup). Remove the saucepan from heat and stir in honey and Dijon mustard. Set aside 1/2 cup for serving.
Brush hens with apple glaze and bake until a thermometer reads 170 degrees, about 25 to 35 minutes longer, basting with pan juices occasionally (if hens brown too quickly, cover the pan loosely with foil).
Remove hens from oven, tent with foil, and let stand 10 minutes. Serve the hens with stuffing, passing the reserved apple glaze separately.
Cornish hens: A type of small chicken that is usually 2 pounds or less. For even cooking, buy hens that are similar in size and weight, about 1-1/4 to 1-1/2 pounds (20 to 24 ounces) each. Thaw frozen hens in a bowl of cold water for 1-2 hours (change the water every 30 minutes) or in the refrigerator for 1-2 days. Place them on a tray to catch any juices that leak from the packaging. Never leave frozen poultry out at room temperature or use warm water to thaw.
Herbs: Fresh herbs taste the best in this stuffing, but dried work too. I rarely find fresh marjoram and almost always substitute dried.
French bread: You can also use brioche, challah, or Italian bread. Dry the bread up to 3 days in advance (keep it covered with a dry kitchen towel on counter, or slice and dry in a 225-degree oven for 30 to 40 minutes).
Yield: This recipe makes 2 Cornish hens with 4 cups of stuffing (2 very generous servings).
Storage: Store leftovers covered in the refrigerator for up to 4 days.
Make ahead: Assemble the stuffing, cover it with foil, and refrigerate it up to 1 day in advance. When the cornish hens go in the oven, pull it out of the refrigerator and let it sit at room temperature for 1 hour. Keep stuffing tightly covered with foil and bake until mostly heated through, about 25 minutes. Remove foil and bake until crispy edges form, about 10 to 20 minutes longer.