Cook corn in a large pot of boiling salted water until just tender, 4 to 5 minutes. Drain and transfer to a bowl of ice and cold water to chill ears quickly. When cold, cut off kernels and scrape cobs over a bowl to extract juice, then purée in batches in a blender with milk, salt, and pepper. Force purée through a very fine sieve into a bowl (discard solids) and stir in chives.
Cut crab shell with kitchen shears and extract meat, then cut into 1/4-inch pieces.
Serve soup drizzled with chile oil and sprinkled with crabmeat.
