If draining yogurt, line a strainer with cheesecloth, set over a bowl, and add yogurt. Cover and chill overnight to drain.
Cut cucumber in half lengthwise and seed. Peel one half, cut into chunks, and purée in a blender. Finely shred remaining cucumber with a box grater or mandoline and squeeze dry with your hands.
In a bowl, stir together yogurt, all of cucumber, 1/2 tsp. salt, the cilantro, and 1 tsp. lime juice. Add milk or water to thin the texture so it's like loose yogurt. Season to taste with more salt or lime juice if you like.
Make ahead: Up to 2 days, chilled.