Begin by preparing the pork belly: Preheat the oven to 400 degrees. Sprinkle all over with the sugar, salt, and pepper. Put it in a casserole dish and bake for thirty minutes. Turn the heat down to 275 degrees and bake it for forty more minutes. Remove it from the oven and let it chill while you make the rest of the dish.
Next, cook the spinach. Heat a tablespoon of butter in a large skillet and cook the spinach until it's wilted. Remove and set aside. Cover to keep warm.
Wipe out the skillet, and heat two more tablespoons of butter. Over medium heat cook the mushrooms until they're soft, about seven minutes. Add the ginger, garlic, miso, mirin, soy sauce, and sesame oil. Cook for three more minutes.
While the mushrooms cook, cook the noodles in boiling water, according to the package directions. Drain the noodles and rinse them under cold water. Drain them again. Add the noodles to the mushrooms and cover to keep the mixture warm.
Slice the pork thinly, and then sear it on both sides in a hot cast iron skillet. Cook it until it starts to crisp up.
At the same time, melt a tablespoon of butter in a large skillet and fry all four eggs until they're cooked through.
Divide the noodles between four bowls. Top each noodle bowl with pork, spinach, eggs, avocado, carrots, and radishes.