In a large bowl, combine the celeriac and onion and season with salt.
Heat about 1 1/2 cups of olive oil in a large frying pan over medium-high heat. Carefully spoon the celeriac mixture into the pan and fry gently until the celeriac is tender. Remove from the pan with a slotted spoon before it becomes crisp or browned and set aside for a few minutes to cool slightly.
Pour the eggs into the celeriac mixture and stir well, then let sit about 5 minutes.
Heat a tablespoon of olive oil in a 10" frying pan over medium heat. Pour the celeriac mixture into the pan and spread evenly. When the egg has cooked around the edges and browned slightly on the bottom but the inside is still runny, flip the omelet. (An easy way to do this is to place a large dinner plate on top of the pan, flip the pan over so the omelet is on the plate, and then slide it off the plate and back into the pan.) You may have to add another tablespoon of oil to the pan if it has dried up.
Cook until set, 3-4 minutes, then turn off the heat and let rest for a couple more minutes before serving.