Cut the carrot into 1/4-inch (6 mm) rounds.
Make the soup. Heat the dashi over medium heat and season with salt, soy sauce, sake, and mirin. Add the carrot rounds, cover, and keep warm.
Prepare toppings. Grill the mochi in a toaster oven or in a frying pan over the stovetop until nicely golden brown. Blanch the spinach, squeeze out excess water, and cut into 2-inch (5 cm) lengths. Thinly slice the green onions.
Add the udon noodles to the hot soup and cook for a few minutes. Divide the noodles into bowls and top with the carrot, mochi, spinach, and green onions. Sprinkle with red pepper if desired.