Any bread will work here, but this is an especially good way to use up leftover bagels. Pumpernickel or everything bagels are our favorite choices for this stuffing.
Preheat oven to 425 degrees. Arrange bread in a single layer on a rimmed baking sheet and bake until dry and beginning to brown, 7 minutes, stirring halfway through. Meanwhile, in a large skillet, melt butter over medium-high. Add onion, celery, and mushrooms and cook, stirring occasionally, until onion is translucent and mushrooms are golden, about 5 minutes. Season with salt and pepper. Transfer to a medium bowl and stir in bread, thyme, and enough broth to moisten (stuffing should feel moist but not soggy). Season to taste with salt and pepper. Using your hands, mix in egg.
Place hens in a roasting pan and season inside and out with salt and pepper. Fill cavity of each with stuffing; tie legs together and tuck wing tips underneath. Roast, covering any exposed stuffing with foil if it begins to overbrown, until hens are golden brown and juices run clear when pierced between breast and leg (an instant-read thermometer inserted in thickest part of a thigh, avoiding bone, should read 165 degrees), about 40 minutes. Tent hens with foil and let rest 10 minutes before serving.
If you can't get the game hens, get a 3 1/2 pound chicken and increase the cooking time to about 1 hour.