In a large pot, combine chicken broth, onion, garlic, ginger, star anise, cinnamon stick, and peppercorns. Bring to a gentle boil, then simmer uncovered for 25 minutes. Strain the aromatics and return the clear broth to the pot.
While the broth simmers, toast the sesame seeds in a dry frying pan over medium heat until fragrant, stirring occasionally. Set aside.
Melt ½ tablespoon butter in the pan. Sauté the chicken breast until fully cooked, seasoning well. Set aside.
Melt the remaining butter and sauté the carrots, leeks, and mushrooms until softened. Season lightly.
Add the sautéed veggies, broccoli, soy sauce, and chili flakes to the pot. Simmer 10 minutes until the broccoli is tender.
While the vegetables simmer, poach the eggs in gently simmering water until the whites are set and yolks remain soft. Set aside.
Add the dried noodles to the soup and simmer a few minutes until tender.
Stir the cooked chicken into the soup and warm through. Taste and adjust seasoning if needed.
Ladle the soup into bowls. Top each with a poached egg, sliced green onions, and toasted sesame seeds. Serve immediately.