I loved to dig as a boy in Vermont, but I gave up in frost. It's an interesting problem. The best solution may depend on the type of soil, its temperature, and how much ice it contains. In Korea, soldiers sometimes used explosives, sometimes fires, and sometimes an assortment of hand tools.
Thawing will stop when the surrounding ground draws heat away as fast as you apply it. I think very cold ground and ashes might even stop thawing from a fire.
Others have recommended light bulbs. That sounds simple and pretty safe. Most of the heat from a bulb is radiant, so lining the box with aluminum foil would reflect more heat to the soil. I'd use several hundred-watt bulbs with porcelain sockets and wire with high-temperature insulation. A little ventilation may be necessary to keep the electrical stuff from overheating.
A professional solution is a trailer-mounted boiler with hoses to circulate water between the boiler and the hole. One could use a stove, two pots of water, and a big syringe or pump to draw water from the hole for reheating. An infrared thermometer would make it easy to monitor the temperature of the water in the hole.