In a small saucepan over medium heat bring all ingredients, except corn starch, to a boil then reduce the heat to low. Spoon 1 Tbl. corn starch into a small fine mesh sifter. Whisk broth while dusting in corn starch, stirring constantly until well blended. Return gravy to a boil, check for desired thickness, add additional corn starch for thicker gravy. Remove gravy from heat and reserve.
For the egg fu yung...
In a medium bowl, sift flour into beaten eggs while whisking briskly. Whisk in soy sauce and sesame oil, then add in bean sprouts, diced ham, scallions, and black pepper.
Heat a skillet over medium-high heat, add 1 Tbl. canola oil then 1/4th of the mushrooms and water chestnuts, saute for 2-3 minutes. Ladle in about 1/2 cup of egg mixture, gently fry for 2-3 minutes per side until golden brown. Replenish oil between batches and adjust heat as needed.
Serve with brown gravy and sliced scallions.
Notes
Diced ham can be replaced with shrimp or left out for a vegetarian version.