These two special strainers both have a hook just at the edge of the rim so they can be securely hung on the edge of the pan. The Miso Koshi (the small one) is used to strain Miso directly into soup, pushing it through with a pestle. Formerly, this had to be done in a Suribachi with a pestle, working in enough of the hot soup stock to suspend the Miso.
The Men Koshi (Larger one) is used for noodles, and especially noodles for Ramen Soup. It may be used either to cook freshly made noodles in a soup bowl quantity, or to reheat previously cooked noodles, again in soup bowl quantity. In either case, they can be easily drained and the noodles transferred to the soup bowl within seconds.
More on Japanese Kitchen Gear