1. Heat a grill to high (450° to 550°). In a blender, whirl together ginger, garlic, vinegar, sugar, cilantro, oil, salt, and 1/4 cup water until very smooth. Pour into a large bowl; set aside.
2. Arrange mussels evenly on cooking grates. Grill, covered, until half the mussels start to open, about 2 minutes. Starting with mussels that have opened the widest, quickly transfer them to the large bowl, being careful not to spill juices from inside shells, and continuing until remaining mussels open. (Discard any that stay closed.) Cover bowl with foil to keep warm.
3. Grill chiles until slightly charred, turning often, 3 minutes. Slice into very thin rings. Grill cucumbers on cut side just until marks appear, about 3 minutes.
4. Set 2 cucumber halves in each of 6 large, shallow bowls. Spoon mussels and cilantro bath over cucumbers. Sprinkle with radishes and chiles.
*Corno di Toro chiles, often called "bull's horn," are long, sweet Italian peppers sometimes sold at farmers' markets.