Put a large, deep skillet over medium-high heat. When it’s hot, add half the oil, swirl it around, and immediately add half the garlic and ginger. Cook for 15 seconds, stirring, then add the broccoli, mushrooms and all but a sprinkling of the scallions.
Raise heat to high, and cook, stirring, until mushrooms release their water and broccoli is bright green and beginning to brown, 3 to 5 minutes.
Sprinkle with salt; add 1-cup water. Stir and cook until almost all liquid evaporates and broccoli is almost tender, another minute or two more, then transfer to a plate.
Turn heat to medium, add remaining oil, then remaining garlic and ginger. Stir, then add chicken and turn heat to high. Cook, stirring occasionally, until chicken has lost its pink color, three to five minutes.
Turn heat to medium. Return broccoli, mushrooms and juices to the pan, and stir. Add soy sauce, sprinkle with more salt and some pepper; add a little more water if mixture is dry. Raise heat to high and cook, stirring occasionally, until liquid is reduced slightly and you’ve scraped up all the bits of chicken. Taste and adjust seasoning, garnish with remaining scallion and serve.