Grate the cucumber on the large holes of a box grater, skin on. Toss with 1/2 tsp sea salt and place in a fine-mesh sieve over a bowl. Let drain for 30 minutes at room temperature.
Transfer the cucumber to a clean kitchen towel and squeeze firmly over the sink until no more liquid drips out.
Mince the garlic finely. Optionally, mash it with a pinch of salt using the flat of a knife into a rough paste.
In a medium bowl, combine the strained Greek yogurt, squeezed cucumber, garlic, lemon juice, and 1/2 tsp sea salt. Stir until fully combined. Taste and adjust salt if needed.
Fold in the chopped fresh dill. Drizzle in the olive oil and stir gently to incorporate.
Taste and adjust with lemon juice, salt, or dill. Transfer to a serving bowl, drizzle with olive oil, and garnish with fresh dill sprigs. Refrigerate for at least 30 minutes before serving.
