In a large pot, bring 6 cups of water to a boil. Add noodles and cook for 3 minutes. Drain noodles and set aside. Rinse pot.

In the same pot over medium-high heat, bring chicken broth, ginger, garlic, and low sodium soy sauce to a simmer. Add carrots and all but 2 tablespoons scallions, and cook until vegetables are tender, about 3 minutes. Use a slotted spoon to remove ginger and garlic.

In a medium bowl, lightly beat together eggs, sesame oil, and salt. Gently stir broth, then pour egg mixture into pot. Let sit undisturbed until eggs are cooked, 40 to 60 seconds. Stir with a wooden spoon to break up eggs. Return noodles to pot and cook for 1 minute.

Divide soup among 6 bowls and garnish with remaining scallions. Serve immediately.