- Preheat oven to 400F.
- In a large bowl, beat egg. Stir in flour and broth. Add oats and stir to form ball. Transfer to a lightly floured surface and roll out to 1/2-inch thickness.
- With a donut cutter, cut dough into shapes and transfer to a parchment-lined baking sheet. (TIP: If you don’t have a donut cutter, use a biscuit cutter to cut shape, then use a small circle cutter to cut out center; a donut pan will not work for this recipe.)
- Bake until browned, 10 to 15 minutes. Remove from oven and let cool completely.
- Prepare frosting: In a medium saucepan, heat frosting ingredients on medium-low, stirring constantly, until carob melts.
- Dip tops of donuts into frosting; let cool. Store in refrigerator for up to 1 week or freeze for up to 3 months.