►Cut cheese into 4 pieces, leaving rind on (it adds extra flavor). Add cheese and 5 quarts water to a heavy bottom stockpot set over low heat. Stir frequently to prevent cheese from sticking to bottom. Bring to a simmer (never boil) and cook over low heat for approximately 1 to 1½ hours, until broth has a strong umami flavor and fat has been released into the broth. (Cooking time will vary depending on age of wheel.) If more flavor is needed add miso paste to taste.
►Remove solids and strain broth. Set aside.
►Melt 8 tablespoons butter in a skillet over medium heat and whisk in miso. Remove corn from cob. Add corn to skillet and cook over low heat for 2 minutes. Set aside.
►Melt remaining butter in a skillet set over low heat. Add mushrooms and cook until golden, about 3 minutes. Set aside.
►Cook noodles according to package instructions. Set aside.
►Bring a large pot of water to a boil. Meanwhile, prepare an ice bath. Add bok choy and blanch for 1 minute, then transfer to ice bath to stop cooking. Drain bok choy and set aside.
►Divide noodles among 4 bowls and top with corn, bok choy, and mushrooms. Add broth to cover. Garnish with bean sprouts and chopped scallions.