Start by hard boiling the eggs. Place them in a pot, cover with water, and bring to a boil. Once boiling, cover the pot and remove it from heat. Let the eggs sit for 10-12 minutes before transferring them to an ice bath to cool. Once cooled, peel and dice the eggs.
Using a grater, peel and grate the cucumber. Place the grated cucumber in a clean kitchen towel and squeeze out excess moisture.
In a large mixing bowl, combine the diced eggs, grated cucumber, Greek yogurt, pressed garlic, olive oil, and diced celery. Gently mix until all ingredients are well combined.
Add salt and pepper to the mixture according to your preference. Taste and adjust the seasoning if necessary.
For the best flavor, cover the bowl with plastic wrap and refrigerate the Tzatziki Egg Salad for at least 30 minutes before serving.