Use a box grater to manually grate the cucumber or you can use a small food processor to finely chop the cucumbers as I do in this recipe. Toss the grated cucumbers with 1/2 teaspoon kosher salt. Spoon the grated cucumber into a cheese cloth or a double thickness napkin and squeeze dry. (There will be a lot of liquid.)
In one large mixing bowl, place the garlic with remaining 1/2 teaspoon salt, white vinegar, and extra virgin olive oil. Mix to combine.
Add the grated cucumber to the bowl with the garlic mixture. Stir in the yogurt, and a pinch of white or black pepper. Stir to combine well.
Cover and refrigerate for a bit (anywhere from 30 minutes to a couple hours before serving). This will help thicken the sauce and give it the best texture, while allowing the flavors to meld.
When ready to serve, stir the tzatziki sauce to refresh and transfer to serving bowl, drizzle with more extra virgin olive oil, if you like. Serve with your favorite veggies, pita chips or wedges.