Grate the cucumber using a box grater or finely chop it in a small food processor. Toss the grated cucumber with half of the kosher salt, then place it into a cheesecloth or double-layered napkin and squeeze out as much liquid as possible.
In a large mixing bowl, combine the garlic with the remaining kosher salt, white vinegar, and extra virgin olive oil, stirring gently to blend the flavors.
Add the drained cucumber to the garlic mixture and mix in the Greek yogurt, a pinch of ground white pepper, and fresh herbs if you choose to use them. Stir everything thoroughly to create a smooth consistency.
Cover the bowl and refrigerate the tzatziki for at least 30 minutes, or up to 2 hours, to allow the sauce to thicken and the flavors to develop.
Before serving, give the sauce a good stir to refresh it, then transfer to a serving bowl. Optionally, drizzle additional olive oil on top and serve with warm pita bread and sliced vegetables.