Grate the cucumber (I use the thicker grate of a box grater), place in a fine mesh sieve, and sprinkle 1/2 teaspoon of the salt. Toss to distribute the salt, and allow to sit for 10 to 20 minutes. Squeeze the excess liquid from the grated cucumber with a clean tea towel. The cucumber will release A LOT of moisture, and removing this will prevent your sauce from becoming watery as it sits. Transfer to a mixing bowl.
Mince the garlic and sprinkle the remaining 1/4 teaspoon salt over top. With the side of a chef’s knife or other wide knife, mash the garlic and salt into a paste. If the mixture isn’t totally smooth, that’s okay.
To the drained cucumber, add the yogurt, garlic paste, oil, vinegar, dill, and pepper. Stir to combine, and then cover and refrigerate. If possible, allow the tzatziki to rest for several hours or even overnight. (The flavors meld and improve, although it’s still tasty when served immediately.)
For a salty kick, you may stir in a 1/2 cup or so of crumbled feta or, when serving as a dip, sprinkle some crumbled feta overtop. A drizzle of olive oil also adds a nice finishing touch when serving as a dip.
Serve the tzatziki