1. Bring the chicken and mushroom broths to a boil in a medium saucepan over medium-high. Season with salt and pepper.
2. Reduce the heat to medium and stir in the mushrooms. Continue cooking for 5 minutes, or until the mushrooms have shrunk to about half their size.
3. Vigorously whisk the egg in a measuring cup that has a spout. Hold the measuring cup with the egg about 12 inches or so above the saucepan of simmering soup and – very slowly – drizzle it into the hot soup, stirring the soup in a circular motion as you go. The egg with immediately cook and form thin ribbons. Don’t worry if the soup becomes cloudy, this is ok because it is thickening up a bit. You want to see this, especially in egg drop soup without cornstarch.
4. Add the scallions to the soup and give it a final stir to mix everything together.
5. Ladle into bowls. Drizzle each bowl with a little sesame oil and a sprinkle of red pepper flakes for added oomph. Serve right away while the soup is piping hot.