Preheat the oven to 350 degrees. Line a baking sheet with parchment paper.
Cook the chicken breast in boiling water for 4-5 minutes, or until cooked through. Cut into pieces and set aside to cool.
Combine the leftover cooked brown rice, cheddar cheese, and cooled chicken in a food processor; pulse until finely minced.
Add the flour to a large bowl and add the chicken broth and whisked egg; mix until nearly combined.
Add the chicken mixture and mix until well combined, adding a bit more broth if needed.
Dust the counter with flour and roll out the dough to approximately ¼″ thickness, and use a dog cookie cutter to cut out treats.
Place on the prepared baking sheet an inch apart. Place into the oven and bake for 25-30 minutes, or until golden brown and crisp.
Allow them to cool on a wire cooling rack completely before storing them in an airtight container in the refrigerator for 1-2 weeks, or in the freezer for a few months.