Steam broccoli florets in the microwave for 2-3 minutes or until tender. You can also do this step on the stove if preferred.
Shred carrots using a regular grater or the shredding attachment on your food processor.
Place shredded carrots, steamed broccoli, egg, 1 cup of the breadcrumbs, chickpeas, cheese, garlic powder, onion powder, salt (if using) and black pepper in a food processor and pulse for about 15 seconds or until well combined. You may need to scrape down the sides with a spatula to ensure all ingredients are incorporated. The mixture should form easily into a ball.
Scoop about a tablespoon of the mixture using a cookie scoop or regular spoon and form into balls. Flatten slightly into round patties about 1/2-inch thick. I got about 27.
Place remaining 1/2 cup breadcrumbs in a shallow dish. Coat each veggie nugget with breadcrumbs.
Heat oil in a skillet over medium high heat until it sizzles when sprinkled with water droplets.
Arrange nuggets in skillet and cook for about 4-6 minutes each side, or until golden brown and crispy. You may need to do this in 2-3 batches depending on the size of your skillet.
Serve with ketchup, ranch, tzatziki, honey-mustard sauce or other preferred dip.