Heat dashi stock, chicken, carrots, soy sauce, mirin, sugar, and salt in a pot over medium heat. Simmer until chicken is no longer pink in the center, 5 to 7 minutes. Add tofu, mushrooms, and bok choy, stirring until vegetables are tender, about 5 minutes more.
Stir udon noodles into broth and simmer until tender, 3 to 4 minutes. Add leeks and crack eggs into soup; simmer until eggs are slightly firm, about 5 minutes more.
