Preheat the oven to 375°F (190°C) and line a baking sheet with parchment paper.
Cook the broccoli: Add florets to boiling water and boil for 3 minutes. Drain well.
Blend the mixture: In a food processor, combine broccoli, cheddar, breadcrumbs, garlic powder, onion powder, salt (if using), and egg. Pulse until combined but still a little grainy.
Shape into tots: Scoop tablespoon-sized portions and roll into small tot shapes.
Bake until golden: Place tots on the baking sheet and bake for 10–25 minutes, flipping halfway, until crispy and golden brown.