Preheat oven to 350 degrees F (175 degrees C). Prick sweet potato several times with a fork.
Heat sweet potato in a microwave on high until tender, about 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 ungreased baking sheet.
Bake until crisp, 35 to 45 minutes. Cool in the pans for 10 minutes before removing to cool completely on a wire rack.
