Cut garlic head in half through the middle to expose centers of cloves. Place garlic in slow cooker. Trim white parts from green onions, and add white parts to slow cooker along with kombu and ginger. Finely chop green tops of green onions, and set aside.
Add tamari, mirin, cane juice (if using), and 8 cups water to slow cooker. Cover, and cook on low for 4 to 8 hours. Strain, and discard solids. Return broth to slow cooker, add mushrooms and carrots, cover, and cook 1 hour more, or until vegetables are tender.
Divide noodles among 4 large bowls. Top with snow peas, then ladle broth over top. Garnish with chopped green onions and, if using, oil and seeds.