There are a few glaciers here in Austria that are open most of the year (if not all year round). Kaprun is one of the more famous ones. I'm sure there are ones in Switzerland and possibly France that would be open all year too.
From what I have heard they are generally quite flat compared to normal resorts. Similarly clothing is a problem and you either have to strip down and risk hurting yourself worse if you fall or staying wrapped up and sweating like a pig.