Place the yogurt in a cheesecloth or paper towel-lined sieve and set it over a bowl. Grate the cucumber and toss it with 1 tablespoon of kosher salt; place it in another sieve, and set it over another bowl. Place both bowls in the refrigerator for 3 to 4 hours so the yogurt and cucumber can drain.
Transfer the thickened yogurt to a large bowl. Squeeze as much liquid from the cucumber as you can and add the cucumber to the yogurt. Mix in the sour cream, vinegar, lemon juice, olive oil, garlic, dill, 1/2 teaspoon salt, and pepper. You can serve it immediately, but I prefer to allow the tzatziki to sit in the refrigerator for a few hours for the flavors to blend. Serve chilled or at room temperature.