Grate the cucumber on the coarse side of a box grater into a bowl, skin on or off, whichever you prefer.
Toss the grated cucumber with 1/2 tsp salt and tip it into a fine-mesh sieve set over a bowl. Let it drain for 10 minutes.
Wrap the drained cucumber in a clean kitchen towel or cheesecloth and squeeze hard over the sink until no more liquid comes out. This step is what keeps the sauce thick instead of watery.
In a mixing bowl, combine the Greek yogurt, squeezed cucumber, grated garlic, chopped dill, olive oil, and vinegar or lemon juice.
Stir everything together until smooth. Taste and add more salt if needed, keeping in mind the yogurt may already be a little salty.
Cover the bowl and chill in the fridge for at least 30 minutes, so the garlic and dill have time to settle into the yogurt.
Give the tzatziki a quick stir before serving. Drizzle with a little more olive oil and a sprinkle of dill on top.