Start the coals or heat the gas grill for hot direct cooking. Make sure the grates are clean.
Trim the root ends and remove any discolored leaves from the radishes. Rinse, pat them dry with paper towels, and toss with the oil until completely coated, including the greens.
Put the radishes on the grill directly over the fire. Close the lid and cook, turning once, until they warm through and char in places, 2 to 3 minutes per side. Transfer to a platter and serve with the butter and salt in small bowls for dipping.