In a small bowl, combine the ground flax, and water. Whisk together and allow to sit for 10 minutes.
While the flax eggs are setting, combine the matzo meal, salt, and pepper. Once the flax eggs have reached a gel-like consistency, add them to the matzo meal bowl, and stir to combine. Add the olive oil and the club soda. Fold together to combine. The mixture shouldn't be dry, but it also shouldn't be too wet. It should have a paste-like consistency. If it's too dry, add an additional tablespoon of club soda. Refrigerate for 30 minutes.
While the matzo meal is cooling, prepare the vegetables. Mince the garlic, peel and chop the carrots, and dice the celery and onion.
Add 1 tablespoon of olive oil to a heavy bottom pan over medium heat. Add the garlic, onion, carrots, and celery. Sauté until onions are translucent, about 3-5 minutes.
Add the vegetable stock, chopped dill, and black pepper to the pot. Increase the heat, and bring to a boil.
Once the matzo meal mixture has chilled and stiffened. Use a cookie scoop, to scoop the matzo meal and form it into balls. Add each ball to the broth. Once all of the balls have been added, cover with a lid, reduce the heat, and simmer for 20 minutes.
When ready to serve, add three matzo balls to each bowl of soup with the broth and vegetables. Top with fresh dill and serve.