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