If a mouse obtained a throwable item from an Event or the Village; such as a ball, wadded up paper, plasma beam, and so on; they can throw it, and it is possible to jump on top of it, hop on top of the ice pillar, fall down to the cheese, and air jump into the hole.
The shaman best way to overcome the pillar is to summon portals. it is recommended to build a bridge before summoning the portals, so the mice won't fall. A good tactic is to build one portal at the bottom of the ramp, and one that lines up with the cheese and hole. Then the mice just slide down the ramp, and since it is angled perfectly, they don't have to move to get the cheese. but this technique is dangerous for inexperienced mice, as they can fall to their deaths. The shaman can also place a box next to the ice that the mice can walljump, which will lead them to the other side. This is also a dangerous technique due to the fact that not all mice might get the cheese and enter the hole.