Wash the cucumber, peel it, cut in half lengthwise, scoop out the seeds, and grate it using a box grater.
Sprinkle 1/4 tsp of fine salt over the grated cucumber, let it sit for 10 minutes, then squeeze out as much liquid as possible using cheesecloth or a fine mesh strainer.
Finely mince or grate the garlic clove, and finely chop the fresh mint and cilantro.
Toast the ground cumin in a dry skillet over medium heat for 30 to 45 seconds until fragrant, then transfer to a small bowl to cool.
In a large mixing bowl, combine the yogurt with the minced garlic, toasted cumin, ground coriander, black pepper, remaining salt, and optional cayenne and chaat masala. Stir thoroughly.
Fold the drained cucumber, chopped cilantro, and mint into the seasoned yogurt gently until just combined.
Cover and refrigerate for at least 15 minutes before serving to allow the flavors to meld.
Before serving, stir gently, transfer to a serving bowl, and garnish with fresh mint sprigs, a dusting of cumin or paprika, and a light drizzle of olive oil if desired.