Step 1: Wash and grate the cucumber using the large holes of a box grater. Place the grated cucumber in a clean kitchen towel or several layers of paper towels.
Step 2: Twist and squeeze the towel firmly over a sink or bowl to extract as much liquid as possible from the grated cucumber. This step is crucial for a thick Tzatziki Sauce.
Step 3: In a medium mixing bowl, combine the strained Greek yogurt, thoroughly squeezed grated cucumber, minced garlic, finely chopped fresh dill, fresh lemon juice (or white wine vinegar), salt, and black pepper.
Step 4: Stir all the ingredients together until well combined. Taste the Tzatziki Sauce and adjust seasonings as needed, adding more salt, pepper, or lemon juice for brightness.
Step 5: Drizzle 1 tablespoon of extra virgin olive oil over the top of the mixture. Stir it in gently.
Step 6: Cover the bowl and refrigerate the Tzatziki Sauce for at least 30 minutes, or ideally 2-3 hours, to allow the flavors to meld and deepen.
Step 7: Before serving, give the Tzatziki Sauce another quick stir. Garnish with an extra sprinkle of fresh dill or a thin slice of cucumber, and a final drizzle of olive oil, if desired.