Preheat oven to 350.
Prick sweet potato several times with a fork. Heat sweet potato in a microwave on high until tender, about 4-6 minutes. Cut potato in half and scoop flesh out of the skin into a bowl; discard skin. Mash potato with a fork or potato masher and transfer about 1 cup to a large bowl. Save any remaining sweet potato for another use.
Mix whole wheat flour, applesauce, and eggs in the large bowl with the sweet potato until a dough forms.
Turn dough out on a well-floured surface and roll dough to about 1/2-inch thick. Cut out shapes using a cookie cutter or cut dough into strips with a pizza cutter.
Arrange cookies on an un-greased baking sheet. Bake until crisp, 30 to 45 minutes. Cool in the pans for 10 minutes before removing to cool completely on a wire rack.