Sauté onion in butter until tender; add the mushroom and ham and cook 3 to 5 more minutes or until the mushrooms are tender. Combine the mushroom mixture, rice, 2 to 4 tablespoons brandy, marjoram, and thyme.
Remove the giblets from hens (reserve for another use). Rinse hens with cold water and pat dry. Sprinkle with salt and pepper. Stuff hens lightly with rice stuffing. Close cavity and secure with toothpicks. Truss and place the hens breast side up in baking pan large enough to fit them.
Bake hens at 350°F for 1 1/4 to 1 1/2 hours, depending on the hens’ size.
Place hens on serving platter.
Combine the sherry and chicken broth; brush over hens. Heat 1/4 cup brandy in a small saucepan over medium heat (do NOT boil). Ignite with a long match and pour over hens after the flame dies down. Serve immediately.