Coarsely grate the English cucumber on the large holes of a box grater into a colander set over the sink. Sprinkle with 0.5 tsp kosher salt and toss. Let stand until it releases liquid, 10–15 minutes.
Squeeze the cucumber by handfuls (or in a clean towel) until very dry; you should have about 1 cup packed drained cucumber.
In a medium bowl, mash the grated garlic with a pinch of the remaining salt using the flat of a knife or a spoon until a paste forms, 15–30 seconds.
Add the Greek yogurt, red wine vinegar, and extra-virgin olive oil to the bowl and whisk until smooth and creamy.
Fold in the drained cucumber, fresh dill, the remaining 0.75 tsp kosher salt, and black pepper until evenly combined.
Taste and adjust seasoning with a touch more vinegar or salt if needed. Cover and refrigerate 30–60 minutes to chill and let the flavors meld.
Stir before serving. The sauce should be thick yet scoopable and glossy; serve cold with grilled meats, vegetables, or bread. Refrigerate leftovers up to 3 days.