Preheat the oven to 400 degrees. Bake the potatoes on a rack for 1 hour or until soft. Allow them to cool slightly.
Cut the potatoes in 1/2, scoop out the pulp, and place in a mixing bowl, setting the skins aside. Add the butter, salt, black pepper, and cayenne. Mash well, adding enough milk to make a fairly stiff mixture. Add the beaten egg yolks and stir well.
Spoon the mixture into the potato skins and top with shredded cheese. Place on a cookie sheet and bake for 20 minutes, until well browned.
Recipe courtesy Junior League Celebration Cookbook