Grate the cucumber aon the large holes of a box grater into another bowl. Sprinkle with the 2 tsp salt and rub into the cucumber with your hands. Set aside 20 minutes, then squeeze to express as much liquid as possible.
Smash the garlic. Sprinkle with salt and with the flat side of a large knife, mash and smear the mixture into a coarse paste.
Stir cucumber, garlic, EVOO, lemon juice, and mint into the yogurt.
Refridgerate for at least 1 hour before serving.