In a large pot, combine garlic, broth, mushrooms, carrots, hoisin sauce, ginger, soy sauce and rice vinegar. Cover and bring to a boil over high heat. Reduce heat to low and simmer for 5 minutes.
Increase heat to high and return to a boil. Add red pepper and sugar peas; cook uncovered for 3 minutes or until tender-crisp. Remove pot from heat. With a slotted spoon, divide vegetables among 4 soup bowls. Stir red chili and sesame oil into pot. Return pot to heat and keep hot.
Add noodles and chicken to bowls, equally dividing. Pour broth over top. Top with eggs and green onions.
Soft Cook Eggs: Put eggs in a saucepan. Add cold water to cover eggs by 1 inch (2.5 cm). Cover and bring to a boil over high heat. Immediately remove from heat. Let stand for 4 minutes. Rinse under cold water.
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