Cold soups may need an extra pinch of salt or squeeze of lime, so be sure to double-check the seasoning after your soup is well chilled.
Shred cucumbers using the large holes of a box grater. Whisk together yogurt, stock, and half-and-half in a large bowl until smooth. Stir in shredded cucumber, green onions, mint, cilantro, jalapeno, ginger, 3 tablespoons lime juice, 1 1/2 teaspoons salt, and 1/2 teaspoon pepper.
Cover and chill 4 to 12 hours. Stir soup well, and, if desired, stir in remaining 1 teaspoon salt, 1/4 teaspoon black pepper, and 2 teaspoons lime juice. Top with watermelon.