Place mushrooms in a bowl and cover with warm water; let soak, about 15 minutes. Drain and squeeze out water from mushrooms; slice.
Set oven rack about 6 inches from the heat source and preheat the oven's broiler on high. Place mochi on a baking sheet.
Broil mochi, turning frequently, until inflated, crisp, and slightly browned, about 5 minutes. Cut into 1-inch strips.
Combine water and dashi-no-moto in a large saucepan; bring to a boil. Reduce heat and simmer until dashi is dissolved, 2 to 3 minutes. Add chicken, soy sauce, and salt; simmer until chicken is no longer pink in the center, about 15 minutes.
Mix shrimp, mushrooms, spinach, carrots, daikon radish, and bamboo shoots into chicken mixture; simmer until vegetables are tender and shrimp is cooked through, about 10 minutes.
Divide mochi strips among 4 bowls and top with soup. Garnish soup with kamaboko slices.