Preheat oven to 325°F. Line a baking sheet with parchment paper.
In a mixing bowl, combine pumpkin puree, peanut butter, and egg. Stir until smooth.
Gradually add oat flour and mix until a firm, slightly tacky dough forms. If too dry, add water 1 tablespoon at a time.
Place dough between two sheets of parchment paper and roll to 1/4-inch thickness.
Remove top parchment and cut dough into long strips or chew sticks with a knife or pizza cutter.
Bake for 30 minutes, then flip each piece and bake another 15 minutes.
Turn off oven and leave chews inside with door slightly open for 1-2 hours to dry and firm up.
Let cool completely before serving. Store in an airtight container in the refrigerator for up to 2 weeks or freeze for longer storage.