Ok, now I have definitely had that issue before. Basically exactly how you described. I replaced only the radiator and had no more issues. Your thermostat is probably ok, but they are so cheap I wouldn't recommend not replacing it also. I think the water in the rad turned to straight steam because it's not under any pressure if it's steaming out like that, plus you may have needed more than a gallon if the water that normally sits in the block was also gone. With only a partial fill, it will turn to steam faster and steam out of the gap quicker.

95 Sportage w/welded and 5.89 geared dana 44's on 38.5" TSL's.- SADLY SOLD
CURRENTLY - 2000 sportage, 5.38 gears, welded diffs, 35" tires, 5.5" lift