Preheat Oven to 350°F degrees. If making a Liver Birthday Cake, line an 8×8 baking pan with parchment paper. If making individual treats, line a rimmed cookie sheet with parchment paper.
In a food processor, add oats and pulse to grind the oats into flour. Add liver and egg on top of the blended oats, and blend until well combined. It will form a thick liquid that looks somewhat like a milkshake.
For a birthday cake: Pour the batter into a parchment-lined 8×8 pan and spread evenly. Bake for 25-30 minutes or until golden brown. For dog treats: Spread the batter onto a parchment-lined baking sheet and smooth it out to about ¼-inch thickness. Bake for 20-25 minutes, then cut into small squares.
Allow to cool completely, and then cut into desired pieces. For a birthday cake: Cut into desired shape using a cookie cutter. If making a multi-layer cake, you can use natural peanut butter, mashed sweet potatoes, or plain Greek yogurt to either frost the cake or hold layers together.For dog treats: Cut into 1-inch squares with a pizza cutter.
Store leftover cake or treats in an airtight container in the refrigerator for up to 5 days. For longer storage, freeze individual portions in freezer bags and thaw as needed.