Pre-heat oven to 375. Brush chicken breasts with olive oil on both sides and sprinkle with salt and pepper. Roast in oven until golden brown, about 20 minutes.
While chicken is roasting, begin heating sesame oil in a large soup-pot. Add ginger and chile paste and heat until fragrant, about 30 seconds. Stir in chicken broth, soy sauce, hoisin and rice vinegar plus a pinch of salt and pepper. Bring to a boil then reduce to simmer.
When chicken is cooked, remove from oven and let cool enough to handle. Use two forks or your fingers to shred chicken. Add shredded cooked chicken and mushrooms to the pot and return to a boil. Reduce heat to low and simmer for 10 minutes.
Divide cooked ramen noodles between four bowls then ladel the chicken ramen on top of each. Garnish with optional toppings to serve.