Harder than it looks, a lot of the floor under the cheese is fake. Either wait for the Shaman to build over the fake holes, or fall down, grab the cheese, and perform a midair jump onto the mouse hole platform. Another method is to wait for some of mice to jump down, then see which cheese have floors and which don't.
Build planks under the cheese, since the floor under it is fake. Albeit most mice are either dead or finished by the time you're able you to something, you might as well help anyway. First, drop a big plank directly beneath you. Jump off the platform and land on the big plank. From there, you can build more planks, such as shown below. Avoid attaching planks directly to the big plank you dropped, as this will tip the plank over.