Grate cucumber using a box grater or finely chop in a food processor.
Toss grated cucumber with 1/2 teaspoon salt in a bowl.
Transfer cucumber to cheesecloth and squeeze out excess moisture.
In a mixing bowl, combine garlic, remaining salt, vinegar, and olive oil.
Add drained cucumber to the garlic mixture and stir.
Fold in Greek yogurt and white pepper. Add herbs if using.
Cover and refrigerate for at least 30 minutes before serving.
Stir before serving and drizzle with additional olive oil if desired.