Preheat the oven to 350F. Place your leftover turkey and veggies in the bowl of a food processor and add the egg and enough gravy and/or stock to process it into a mixture with the texture of thick, chunky sauce.
Pour the mixture into a bowl and add enough flour to make a workable dough. Roll it into balls and press down with the tines of a fork on a cookie sheet, or roll or pat out on a floured surface and cut into biscuit shapes. Place on a parchment-lined sheet and bake for 20 minutes, or until golden. To get them nice and hard, turn the oven off but leave them inside as the oven cools—when they're completely dried out, they'll last longer. If they still have some moisture to them, keep them in a sealed container in the fridge.
Makes about 3 dozen treats, depending on their size.