Make the sauce: whisk together the soy sauce, 2 tablespoons toasted sesame oil, hoisin sauce, and chili flakes in a small bowl; set aside.
Heat 1 tablespoon oil in a large skillet over medium heat. Season beaten eggs with a pinch of salt and pepper, add to the skillet, and cook until edges set, about 1–2 minutes; roughly scramble, transfer to a plate, and set aside.
Increase heat to medium-high and add 2 tablespoons oil. When shimmering, add mushrooms and cook, stirring occasionally, until edges brown, about 5 minutes. Season with salt and a large pinch of black pepper.
Stir in chopped green onions, grated ginger, minced garlic, and chopped bok choy (or spinach); cook 2–3 minutes until greens wilt. Remove vegetables and mushrooms to the plate with the eggs.
Return skillet to medium-high and add remaining 2 tablespoons oil. Add cubed chicken and cook, stirring occasionally, until browned and cooked through, about 5 minutes.
Stir 2 tablespoons of the prepared sauce into the chicken and cook 1–2 minutes. Add the day-old rice and cook, stirring and pressing up any toasted bits, until rice is heated and slightly toasted, about 4–5 minutes.
Pour in the remaining sauce, toss to combine, and cook another 1–2 minutes. Stir the mushroom-and-greens and the scrambled eggs back into the pan until evenly distributed; remove from heat.
Fold in chopped basil or cilantro, taste and adjust seasoning with salt and pepper if needed, then divide among bowls and top with extra sliced green onions.