Make the stock. Bring water to a boil. Simmer kombu and dried shiitake mushrooms for about an hour. Strain out the vegetables and retain the stock.
Meanwhile, prepare the squash. Preheat the oven to 375 degrees. Cut the kabocha in half. Use a large spoon to scoop out the seeds. Cut the skin off the squash using a large chef’s knife. Alternately, try using a peeler to remove the skin.
Cut the squash into three-quarter inch thick slices. Toss the squash in a large bowl with the vegetable oil and 1 tablespoon of the ginger. Line up the squash on a baking tray lined with parchment. Bake for about 20 minutes until nicely browned, turning a few times to ensure even browning. Remove from the oven and set aside until needed.
Simmer the broth for about 10–15 minutes with minced ginger, shiitake mushrooms, tamari, and mirin. Add udon noodles, baby bok choy, Asian pear, and scallions, and simmer for another 3–4 minutes.
Remove about half a cup of broth from the stock and mix it with the red miso until it is a smooth paste. Turn the heat off the broth and stir in the miso.
Serve in wide bowls and top with squash slices and a drizzle of sesame oil.