Heat 1 tablespoon vegetable oil in a wok or skillet over medium-high until very hot. First, add the onions and cook for 5 minutes until beginning to brown. Next, add the sliced shiitake caps and cook, stirring regularly until well-browned, about 6-7 minutes more.
Add the green beans and cook until bright green and tender-crisp. Season very lightly with salt. Transfer the veggies to a bowl.
Add the chicken to the hot skillet and cook without moving until beginning to brown and crisp up in spots, about 5 minutes.
Turn the heat to high and once very, very hot, add the sauce, scraping up any browned bits stuck to the bottom. Reduce heat and cook for 2-3 minutes more. Return the vegetables back to the pot and toss to coat. Remove from heat.