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. Add the beaten eggs (if using), season with a pinch of salt and pepper, and cook until the edges set, about 1–2 minutes; roughly scramble and transfer to a plate.
Increase heat to medium-high and add 2 tablespoons oil. When the oil shimmers, add the mushrooms and cook, stirring occasionally, until edges brown, about 5 minutes. Season with salt and a large pinch of pepper.
Stir in the green onions, grated ginger, garlic, and chopped bok choy (or spinach) and cook another 2–3 minutes until greens are wilted and mushrooms are caramelized. Transfer mushrooms and greens to the plate with the eggs.
Return skillet to medium-high heat and add the remaining 2 tablespoons oil. Add the 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 cooked day-old rice and cook, stirring occasionally, until the rice is heated through and slightly toasted, about 4–5 minutes.
Add the remaining sauce and toss to combine, cooking another 1–2 minutes. Stir the reserved mushrooms, greens, and scrambled eggs back into the rice mixture until evenly distributed.
Remove from heat, stir in the chopped basil or cilantro, then divide among bowls and top with additional green onions if desired.
