Prep the Brussels sprouts: Trim off the steam and any discolored outer leaves from the Brussels sprouts. Then boil them in well salted water until still crisp tender, about 6 or 7 minutes for smaller ones, perhaps 10 minutes for larger ones. Drain them and let them cool.
While the sprouts are cooling, prepare the batter. Whisk together the flour and water, then mix in the grated cheese, salt and olive oil. Separate the eggs. Whisk the yolks into the batter. Whip the whites until they form soft peaks (you'll want an electric whisk for this until you have really strong arms) and fold them gingerly into the batter.
Heat the oil in a deep fryer until it's moderately hot, say 180C/360F. Mix the Brussels sprouts in the batter until they are all well coated. One by one, pick the sprouts out of the batter with some tongs, shake them gently to shed the excess batter, and plop them gingerly into the hot oil, taking care to avoid splashing the hot oil. Deep fry the sprouts for about 5 minutes, or until they have browned to your liking.
Serve while still warm, sprinkled generously with salt.