1. Freeze beef 10 minutes; cut across grain into 1/8-inch-thick slices.
2. Cook noodles according to package directions. Drain and rinse with cold water; drain.
3. Place onion and next 5 ingredients in a large saucepan; cook over medium-high heat 5 minutes, stirring frequently. Add broth and 3 cups water; bring to a boil. Strain broth mixture through a fine sieve over a bowl; discard solids. Return broth to pan. Add soy sauce, sugar, and fish sauce; bring to a boil. Add bok choy and snow peas; simmer 4 minutes or until peas are crisp-tender and bok choy wilts.
4. Arrange 1/2 cup noodles in each of 4 large bowls. Divide raw beef and chile slices evenly among bowls. Ladle about 1 2/3 cups hot soup over each serving (broth will cook beef). Top each serving with 1/4 cup bean sprouts, 1 tablespoon basil, and 1 tablespoon mint. Serve with lime wedges.
Freezing the beef for just 10 minutes makes it easier to cut into thin 1/8-inch slices, so it's worth taking the time to complete that step.