Grate the cucumber on the large holes of a box grater. Toss with 0.5 teaspoon of the salt and place in a fine sieve set over a bowl; let drain 10–15 minutes, then squeeze firmly by hand or in a clean towel until very dry.
In a small bowl or mortar, mash the garlic with the remaining 0.5 teaspoon salt to a smooth paste. Stir in the white wine vinegar and extra-virgin olive oil until lightly emulsified, about 30 seconds.
Place the Greek yogurt in a mixing bowl (if not already very thick, strain it in a cheesecloth-lined sieve for 30 minutes first). Stir in the garlic–vinegar–oil mixture until smooth.
Fold in the well-squeezed cucumber until evenly distributed. Taste and, if desired, adjust the acidity with a few drops more vinegar.
Cover and chill 1–2 hours for the flavors to meld. Stir before serving and serve cold.