In the bowl of a food processor, combine carrot, broccoli, parsnip, beet, and garlic until thoroughly chopped, stopping to scrape down the sides as necessary. Add the eggs, seasonings, and breadcrumbs and continue to process until a thick paste forms.
Spread the mixture out on a parchment lined baking sheet so it is about ¼ inch thick. Cover with plastic wrap and refrigerate 30 minutes or up to overnight.
Preheat the oven to 400ºF. Line a large baking sheet with parchment or a silicone baking mat. Place about ¾ cup breadcrumbs in a shallow bowl.
Use a cookie cutter or small biscuit cutter to cut the mixture into shapes. Gently toss each nugget in the extra breadcrumbs, pressing to adhere. Place the coated nuggets on the prepared baking sheet. Cut out the shapes closely to avoid too much waste. There will still be a lot of leftover veggie mixture. You can carefully reshape this and cut out more nuggets if you’d like, though they may not hold together as well and will likely need to be baked in batches.
Bake the nuggets for 25 minutes, flipping once halfway through.
Serving size and nutrition information is difficult to calculate based on your cutter size and leftover veggie mixture after cutting out the nuggets. I have based the serving numbers on reshaping the dough and the nutrition facts on one nugget. Servings and nutrition facts may vary.