Add the egg to a medium mixing bowl and whisk until lightly beaten.

Stir in the pumpkin puree, 1 tablespoon of oil, and peanut butter (if using) until the mixture is smooth and well combined.

Add the pulsed rolled oats and the flour to the bowl and stir with a sturdy spoon or spatula until a thick, slightly sticky dough forms.

If the dough is too wet to handle, sprinkle in a little more flour; if it is crumbly or dry, add water 1 teaspoon at a time, mixing until the dough holds together.

Lightly flour your work surface and your hands, then turn the dough out onto the surface.

Divide the dough into 2–3 portions, then roll each portion into a log about 1/2 inch thick.

Cut the logs into snack sticks about 4–5 inches long, or shorter for small dogs, reshaping gently as needed.

Preheat a grill pan over medium-low heat and lightly oil it with the remaining teaspoon of oil, wiping away any excess with a paper towel.

Place the snack sticks on the warm grill pan in a single layer without crowding.

Cook for 4–5 minutes on the first side, until firm and lightly browned with grill marks.

Turn each stick and cook for another 4–5 minutes, adjusting the heat as needed to avoid burning.

Continue turning the sticks every few minutes for a total cook time of about 15–20 minutes, until the outside feels set and lightly crisp while the inside is cooked through.

Transfer the cooked snack sticks to a wire rack and cool completely to room temperature before serving to your dog.

Store cooled sticks in an airtight container in the refrigerator for up to 5–7 days.