Heat sesame oil in a 3-quart classic-style donabe over medium. Add garlic, ginger, and white scallion pieces. Cook, stirring constantly, until softened and fragrant, about 2 minutes. Push mixture to 1 side of pot, and add chile bean sauce to empty side. Cook, stirring constantly, until fragrant, about 30 seconds. Stir in sake, incorporating garlic, ginger, and white scallion pieces; cook 1 minute. Whisk together miso and 1/4 cup shiitake dashi in a small bowl until smooth; set aside. Stir soy sauce and remaining 3 3/4 cups shiitake dashi into pot. Cover and bring to a simmer over medium-low. ncover and stir in cabbage, mushrooms, and carrots. Add tofu. Cover and cook until tofu is heated through and vegetables are crisp-tender, 4 to 5 minutes. Stir in miso mixture. Using a spider or slotted spoon, divide vegetables and tofu between 2 serving bowls (about 2 cups each). Ladle about 1 cup of broth into each bowl; reserve remaining broth in donabe. Garnish with ground sesame seeds, chile oil, kurozu, quick-pickled shiitakes, and green scallion pieces. Reheat remaining broth (about 1 1/2 cups) in donabe over medium-low until steaming. Add ramen to broth; let stand until heated through. Serve as second course with hot pot.