Fill a large, heavy-bottomed pot with 2 inches oil, and heat over high heat until a deep-fry thermometer reaches 350 degrees. Working in 3 batches, fry plantains until golden, about 4 minutes, and transfer to a paper-towel-lined plate with a slotted spoon (return oil to 350 degrees between batches). Flatten plantains with a mallet until very thin. Return oil to 350 degrees. In batches, refry plantains until golden brown and crisp, about 3 minutes. Transfer tostones (the plantain chips) to a paper-towel-lined plate, and season with 1/4 teaspoon salt.
Gently combine crabmeat, mayonnaise, avocado, lime juice, chives, and 1/2 teaspoon salt. Divide evenly among tostones, and garnish with radishes. Crab salad can be refrigerated, surface covered with plastic wrap, up to 8 hours. Tostones can be stored at room temperature, uncovered, up to 2 hours.