Preheat oven to 400 degrees F (200 degrees C). Line a baking sheet with parchment paper.
Place 1/2 of the broccoli florets in a food processor; pulse about 20 times until it looks like grains of rice. Repeat with remaining broccoli florets.
Fill a large, shallow saucepan with 1/2 inch water; add 1 teaspoon salt. Bring to a boil. Reduce heat to low and add broccoli rice. Cover and simmer until tender, about 5 minutes. Drain and let cool, about 10 minutes.
Transfer broccoli rice to the center of a clean absorbent dish towel. Squeeze and twist the towel to remove as much of the moisture as possible.
Mix Cheddar cheese, bread crumbs, onion, egg, parsley, garlic, cumin, and black pepper together in a large bowl. Mix in broccoli rice until mixture sticks together when pressed into a ball.
Scoop 1 1/2 tablespoons of mixture with a small ice cream scoop; form into a ball and press into the shape of a tot using oiled hands. Place on the prepared baking sheet. Repeat with remaining mixture.
Drizzle olive oil over broccoli tots. Sprinkle remaining 1 teaspoon salt on top.
Bake in the preheated oven, turning halfway with tongs, until golden brown, about 24 minutes.