Grate the broccoli heads and stems (the food processor grate blade works best). Add grated broccoli to a large bowl, add salt, and toss to thoroughly mix. Allow mixture to sit for at least 20 minutes. This process will draw out excess moisture.
To drain, squeeze handfuls of the mixture over a fine mesh sieve and then transfer to a paper towel. Repeat until you've squeezed all of the mixture, and then use the paper towel to squeeze any remaining liquid out. Transfer dried broccoli back to the bowl.
Add egg and mix well to coat. Add garlic, bread crumbs, and cheese and mix well.
Preheat oven to 400 degrees F and line a large baking sheet with parchment paper.
Use your hands to squeeze tablespoon amounts of the broccoli mixture into bite sized tots. Lay one inch apart on the baking sheet.
Bake in preheated oven for 22-24 minutes, rotating at least once during baking process. Serve warm alone or with your desired dipping sauce.