In a large pot over medium heat, warm the oil. Add the mushrooms and spread out into an even layer. Let cook for 3-5 minutes, undisturbed, or until browned in spots. Stir in the white and light green parts of the scallions and ½ teaspoon salt. Cook for 1 more minute, stirring occasionally.
Add the garlic and red pepper flakes (if using) and cook for 1 more minute, stirring often.
Add water, cabbage/bok choy, soy sauce, ginger, sesame oil, and remaining 1/2 teaspoon salt. Bring to a boil, then simmer rapidly for 8 minutes.
Add the wontons, bring the water back up to a boil, and boil for 2-3 minutes (or according to the package instructions), making sure the wontons are submerged.
With the water boiling, slowly pour in the whisked eggs in a circular motion (see video). Turn off the heat and let it sit until the eggs become mostly cooked, about 30 seconds (the heat of the broth will continue to cook the eggs).
Serve topped with the dark green parts of the scallions. I usually fish out the dumplings and divide them between the bowls, then scoop the broth/veggies and pour that over top.