Whisk eggs, neutral oil, matzo-mixture water, its salt and pepper in a bowl. Stir in matzo meal, cover and refrigerate for 30 minutes, additional to active prep time.
Bring a large pot of water to a gentle simmer and add the tablespoon salt. With wet hands, shape the chilled mixture into 12 loose balls without packing them tightly.
Lower the balls into the water, cover and simmer gently for 30 to 35 minutes until puffed, tender and at least 160°F in the centers.
Meanwhile, heat olive oil in a soup pot and cook carrots and celery for 5 minutes. Add broth, chicken and soup salt and simmer for 20 to 25 minutes until the chicken reaches 165°F.
Remove and shred the chicken, then return it to the broth. Lift the cooked matzo balls from their water with a slotted spoon.
Place two matzo balls in each bowl and ladle the chicken, vegetables and clear broth around them.