Grate cucumbers using the small holes on a box grater. Place grated cucumber in a fine-mesh sieve set over a bowl. Sprinkle with ¼ teaspoon salt and let stand 10 minutes. Wrap cucumber in a clean kitchen towel, and squeeze firmly until excess moisture is removed.
In a large bowl, whisk together yogurt, lemon zest and juice, olive oil, dill, mint, garlic, pepper, and remaining ¾ teaspoon salt until smooth and well combined. Stir in cucumber.
Add chicken, celery, and shallots. Stir until evenly coated. Season to taste with additional salt, pepper, or lemon juice, if needed.
Cover and refrigerate until flavors meld, at least 30 minutes or up to 4 days.
Serve on toasted bread, in pita pockets, lettuce cups, or over salad greens. Drizzle with olive oil, and garnish with dill and black pepper, if desired.
