Preheat oven to 425F, place oven rack in the middle of the oven, and line a baking sheet with parchment paper or a silicone mat.
Make the mixture:
Boil, steam, or microwave the potatoes to the point where a fork punctures through but they are too tough to cook -about 10 minutes in boiling water, 5-6 in the microwave. Drain the potatoes and set aside to cool down slightly.
In a food processor, pulse the broccoli florets until it resembles rice. Transfer into a large mixing bowl.
Transfer the potatoes to the food processor and pulse a few times to break them up into small pieces. You don't want them fully mashed; you want to leave a little texture like real tots. Add them to the riced broccoli bowl.
Add 1 ½ cups of parmesan cheese, the Italian seasoning, and a little salt and pepper to the bowl. With a spatula, fold the ingredients together to combine, making sure no large pieces of potatoes are left.
Form the tater tots:
With your hands, begin forming small balls -smaller than meatballs, larger than traditional tater tots, about 1 tablespoon in size if using a mini cookie scoop.
Roll each ball in the remaining parmesan cheese and place them on top of the lined baking sheet.
Cook:
In the oven, bake them in a preheated oven for 12 minutes until they are golden brown. Remove from the oven and serve warm.
Air fryer: preheat the air fryer for 5 minutes at 375F. Lightly spray the broccoli tots and place them on top of the crisper tray. Cook for 8-10 minutes. This will vary slightly by their size.
Storage:
Refrigerate cooked broccoli tots for up to 4 days. Reheat in the air fryer for 4 minutes at 340F or in the oven at 350F for about 6 minutes. You can freeze unbaked tots for up to 2 months. They thaw out quickly while preheating the oven or air fryer and usually only require 2 additional minutes of cooking time.