I wonder if you could use two sheets of glass, and have a double-paned system. Maybe even with vacuum or some gas that's an even poorer conductor of heat than air in it.

This is one for windows --- unfortunately, even the best such windows have a markedly lower R-factor than the much less expensive typical door construction.

Such refrigerators are available for commercial applications --- where the on-going energy usage is offset by the need for customer access, but it would be a huge impact on one's electric bill.

Bars, restaurants and supermarkets have glass fronted refrigerators and freezers the world over, and yes, most are double glazed.

They're just less efficient and more likely to break. In a commercial environment that's worth it to show off the contents, but the trade off is different in a domestic setting.