Preheat the oven to 350°F and line a baking sheet with parchment paper.

Add the blueberries and water or low-sodium chicken broth to a small saucepan.

Cook over low to medium-low heat, stirring occasionally, until most berries have burst and the mixture looks loose and jammy, about 5 to 7 minutes.

Remove the blueberry mixture from the heat and let it cool to room temperature.

While the blueberries cool, place the rolled oats in a food processor and pulse until they form a coarse flour.

In a mixing bowl, combine the oat flour, mashed banana, and egg, and stir until a soft dough forms.

Lightly flour a work surface with a bit of oat flour and roll the dough out to about 1/4 inch thickness.

Spoon small amounts of the cooled blueberry mixture over the surface of the rolled dough.

Use a butter knife or toothpick to gently drag the blueberry mixture through the dough to create swirled, marbled patterns.

Use a small cookie cutter or knife to cut the dough into bite-size shapes and place them on the prepared baking sheet.

Bake for 18 to 20 minutes, or until the treats are set and lightly golden at the edges.

Let the treats cool completely on a wire rack before serving to your dog.