Preheat the oven to 350 degrees F (175 degrees C).

Prick sweet potato several times with a fork. Cook in the microwave on high until tender, about 6 minutes. Let sit until cool enough to handle.

Cut potato in half and scoop flesh into a bowl; discard skin. Mash potato with a fork or potato masher. Transfer about 1 cup mashed potato to a large bowl, saving any remaining for another use.

Mix whole wheat flour, eggs, and applesauce into sweet potato in the bowl until a dough forms.

Turn dough out onto a well-floured surface and roll dough to about 1/2-inch thickness. Cut out shapes using a cookie cutter or cut dough into strips with a pizza cutter. Arrange cookies on an ungreased baking sheet.