Grate the English cucumber on the large holes of a box grater. Toss with 1/2 teaspoon kosher salt in a fine-mesh strainer and drain for 10 minutes.
Wrap the shreds in a clean lint-free towel or cheesecloth and squeeze over the sink until liquid no longer streams out. The cucumber should remain damp and fluffy.
Stir Greek yogurt, garlic, olive oil, lemon juice, dill, optional mint and black pepper in a medium bowl until smooth.
Break apart the squeezed cucumber and fold it evenly through the yogurt base. The sauce should hold soft ridges from a spoon.
Cover and refrigerate at least 30 minutes so the flavors combine and the yogurt firms slightly.
Stir, taste and add salt or lemon a little at a time. Spoon into a shallow bowl, drizzle lightly with olive oil and serve chilled.
