The shaman should make sure there are no mice currently on the right side of the map, and use a cannon pointing right to blow the blocks off the edge of the map, releasing the cheese so mice can gather it and get to the hole. Multiple cannons may be required to get every block moved. If you are afraid you may kill mice in this process, use space cannons and ghost arrows, even if it takes slower to accomplish.
Rushing is usually the best option, as the map can be completed without a Shaman. Be wary of what the shaman is doing, because a lot of them tend to cannon the tower, which usually leads to at least one completely avoidable death.
It is recommended to do nothing until the rushing mice block the cheese by knocking the blocks on it, then build Ghost items to push the blocks out of the way. Alternatively, it is possible to just cannon the blocks away, but you risk killing the rushing mice in the process. If you have the Big Plank and Restorative skill, you can spawn an upright plank, stablize it, then cannon it into the blocks.