Place the peeled potatoes in a saucepan and fill with cold water. Bring to a boil, and continue boiling until the potatoes are just barely cooked through (important!), about 20 minutes.
Preheat the oven to 425 degrees Fahrenheit, and line a baking sheet with parchment paper.
Grate the potato into a mixing bowl using the largest holes on your grater.
In a small bowl, whisk together the aquafaba and sea salt with a fork. Add about half of this mixture to the grated potato and stir.
Add the shredded Brussels sprouts to the bowl along with the rest of the aquafaba mixture (this is just to ensure the salt and aquafaba are distributed evenly), and stir to combine with the potatoes. Taste a pinch of the mixture and add more salt if desired (it won't taste great at this point because of the raw sprouts, but it's safe to eat).
By hand, form the mixture into small balls or tater tot shapes, squeezing each one gently to compress it a little bit, and place on the prepared baking sheet. Repeat until the mixture is used up. The baking sheet should be large enough that none of the tots are touching. If needed, use a second baking sheet rather than crowding the first one.
Brush the tops of the Brussels sprout tater tots with a very light coating of olive oil. Sprinkle with additional salt if desired. Bake for 15 minutes, then very gently flip over the tater tots. Bake for another 15-20 minutes or until lightly browned and crisp.
Serve immediately. Leftovers can be stored in an airtight container once completely cooled, and are most successful when reheated in the oven or toaster oven.
Notes
For variations: add other seasonings to taste; I especially like a bit of garlic powder and paprika, or Old Bay seasoning.Other varieties of potato are unlikely to be successful in this recipe because of varying starch content.Aquafaba is the cooking liquid from beans. For this recipe I used chickpea aquafaba, and I simply used the liquid from a BPA-free can of chickpeas. You can also use homemade aquafaba from cooking beans, but make sure it is reduced enough to be slightly viscous. It acts as a binder in this recipe. Please note that aquafaba is NOT the same thing as soaking liquid.It should be easy to form the mixture into intact shapes. If they aren't holding together, the potatoes were probably cooked too long. To salvage the tater tots, try adding potato flour/starch or another light flour a little bit at a time, just until the mixture holds together.