In a medium pot over medium heat, add the oil, scallions, mushrooms, carrot, and salt. Cook for about 3-5 minutes, stirring occasionally, or until softened.
Stir in the water and greens (see note). Bring to a boil over high heat, then reduce the heat to low and simmer for 5 minutes.
Soak the wakame in cold water, if using (see note).
In a small-medium bowl, whisk the eggs until smooth.
For a classic egg drop (large curds): skip the tempering step described below and slowly pour the eggs into the soup while slowly stirring the soup.
For a creamier egg drop (smaller curds): to temper the eggs, slowly pour in about ½ cup of the broth into the eggs while whisking constantly. Then slowly pour the eggs into the soup while stirring constantly. The soup should turn a little creamy with smaller egg strands.
Stir the seaweed into the soup (if using), then turn off the heat.
Transfer about ½ cup of water to the same small bowl as the eggs along with the miso. Mash/mix in the miso paste until it’s dissolved. Stir into the pot.
Serve right away topped with more scallions if you’d like.