Slice cucumbers thinly (about 1/8 inch). Place in a colander, toss with 1 teaspoon salt, and leave for 15–20 minutes. Squeeze or press with a clean towel to remove as much liquid as possible.
Combine Greek yogurt, grated garlic, lemon juice, olive oil, salt, pepper, and cumin in a bowl. Stir until smooth. Taste and adjust seasoning before adding cucumber.
Add drained cucumber slices and red onion to the dressing. Fold in most of the dill and mint. Toss gently until everything is evenly coated. Taste and adjust salt or lemon as needed.
Cover and refrigerate for 20–30 minutes before serving. Stir before plating. Transfer to a serving bowl, drizzle with olive oil, and scatter reserved dill and mint on top.