In a large pot, combine the chicken leg quarter, water, halved onion, crushed garlic, and sliced ginger.
Bring to a boil over high heat, then reduce to a simmer and cook for 30 minutes until the chicken is cooked through and the broth is flavorful.
Remove the chicken from the broth and set aside to cool. Strain the broth through a fine sieve into another pot, discarding solids. Return the clear broth to the stove and keep warm.
While the broth simmers, prepare the noodles: in a mixing bowl, combine the all-purpose flour and 1/2 teaspoon salt.
Gradually add 3/4 cup water and mix until a dough forms. Knead the dough on a floured surface for about 5 minutes until smooth and elastic.
Cover the dough with a damp cloth and let it rest for 15 minutes.
Roll out the dough thinly (about 1/8 inch thick) on a floured surface, then cut into thin strips about 1/4 inch wide to make noodles.
Bring a separate pot of water to a boil. Add the noodles and cook for 3-4 minutes until tender but still chewy. Drain and toss with 1 tablespoon vegetable oil to prevent sticking.
Shred the cooled chicken meat, discarding bones and skin.
Add the shredded chicken, julienned carrot, and green onion pieces to the warm broth. Season with 1 tablespoon soy sauce, 1 teaspoon salt, and 1/2 teaspoon black pepper. Simmer for 5 minutes.
Divide the cooked noodles into serving bowls. Ladle the hot chicken broth with vegetables and chicken over the noodles.
Drizzle each bowl with 1 teaspoon toasted sesame oil and sprinkle with toasted sesame seeds before serving.