1. Heat oven to 400 degrees F. Pierce the sweet potato several times with the tines of a fork. Place the sweet potatoes on a rimmed baking sheet lined with foil. Bake until tender, about 45 minutes. Transfer to cooling rack. Reduce oven to 350 degrees F and line a baking sheet with parchment paper.
2. When cool enough to handle, cut the potato lengthwise and use a spoon to carefully scoop out the insides; discarding the skin. Puree the sweet potatoes in a food processor until completely smooth.
3. In a large bowl, whisk together the flour and flaxseed meal. Remove about 1 tablespoons of the egg to a separate cup.
4. Add the sweet potato puree and remaining beaten egg to the dry ingredients and mix until a dough forms.
5. Transfer to a work surface and roll the dough into a 1/2-inch-thick rectangle. Cut mixture using a canine-themed cookie cutter and place on a prepared cookie sheet. Brush with the remaining 1 tablespoon beaten egg. Bake for 25-35 minutes, or until golden brown. Remove to cooling racks and allow to cool completely before storing in an airtight container.