Rinse the fish fillet and pat dry with paper towels.
Cut the fish fillet into bite-size pieces. Toss with the salt, pepper and corn starch. Set aside.
Mix together all the ingredients for the sauce. If using dried mushrooms, add two tablespoons of the soaking liquid.
Cut the broccoli into florets.
Discard the mushroom stems and cut the caps into bite-size pieces.
Heat water (about three inches deep) in a wide shallow pan.
Dump the broccoli florets in the boiling water. Sprinkle in three generous pinches of salt. Cook for a minute.
Drain the broccoli and pour into a bowl of iced water. Leave to cool then drain.
Pour off the water from the pan. Heat the pan and, when dry, pour in enough cooking oil to reach a depth of at least three inches.
When the oil is hot, drop in the seasoned fish, one at a time, and cook until golden. Remove with a slotted spoon and set aside.
Pour off the oil from the pan (you can reuse the oil). Pour in the sauce, heat and allow to boil gently for a minute.
Add the shiitake and quail eggs to the sauce. Cook, tossing the pan, occasionally, for about two minutes.
Add the fish to the mushrooms and quail eggs. Toss around to coat each piece with the sauce. Cook for another minute.
When the sauce has reduced and thickened, turn off the heat. Toss in the broccoli.
Serve immediately with hot rice.