Rub the chicken breasts with the red chili paste, making sure they are evenly coated. Heat 1 tablespoon of sesame oil in a skillet over medium heat. Pan-fry the chicken for 5–6 minutes per side, until cooked through and lightly golden. Remove from the pan, let rest for 5 minutes, then slice into thin strips.
In a large pot, heat the remaining 1 tablespoon of sesame oil over medium heat. Add the garlic and ginger and sauté for 1–2 minutes, until fragrant.
Stir in the soy sauce, rice vinegar, and sriracha. Pour in the chicken broth and bring to a gentle simmer.
Add the mushrooms, shredded carrots, and the chopped bok choy stems (the thicker white parts). Cook for 3–4 minutes, until the vegetables begin to soften.
Add the ramen noodles and the bok choy leaves directly to the broth. Cook according to the noodle package instructions, stirring occasionally, until the noodles are tender and the bok choy is wilted.
Divide the noodles and broth between bowls. Top with the sliced chicken, soft boiled egg halves, green onions, sesame seeds, and red pepper flakes. Serve hot, with extra sriracha if desired.