Grate the cucumber on the large holes of a box grater. Wrap in a clean kitchen towel and wring out as much moisture as possible — this step is non-negotiable for thick tzatziki.
In a bowl, combine Greek yogurt, squeezed cucumber, garlic, dill, mint, olive oil, and vinegar.
Stir well and season with salt. Adjust — more garlic if you like it bold, more lemon if you want brightness.
Cover and refrigerate at least 30 minutes (overnight is even better — the flavors deepen beautifully).
Serve drizzled with a little extra olive oil and a sprig of fresh dill.