In a medium mixing bowl, whisk together 1 tablespoon soy sauce and cornstarch with a fork. Add the turkey, breaking apart the meat and stirring to coat with the sauce. Let marinate 10 minutes.
Meanwhile, chop the mushrooms very finely and mince the garlic. Thinly slice the green onions. If any of your other ingredients are not yet prepped, do it now.
Heat a wok or large, deep sauté pan over high heat. Add the oil and swirl to coat. Add the marinated turkey and cook, breaking apart the meat into small bits with a heatproof spatula, until it's no longer pink and fully cooked through, about 5 minutes. Add the green onions (reserve a small handful for serving) and mushrooms and cook until the mushrooms soften, about 2 additional minutes.
Reduce the heat to medium-low. Add the broccoli coleslaw, carrots, ginger, and garlic. Stir-fry for 2 minutes, until the vegetables are softened.
Add the rice vinegar, chili paste, sesame oil, black pepper, and remaining 2 tablespoons soy sauce, then stir to combine. Continue to cook for 1 additional minute. Taste and add additional soy sauce, hot sauce, or black pepper as desired. Scoop into serving bowls (over brown rice, quinoa, or cauliflower rice if desired). Serve hot, sprinkled with additional green onions.
TO STORE: Store the egg roll filling in an airtight storage container in the refrigerator for up to 4 days. You can store the rice on its own or portion everything together into individual serving containers for a meal-prep approach.
TO REHEAT: Rewarm leftovers in the microwave or in a wok over medium-low heat. I find the leftovers lose some of their zip, so I like to top mine with a few shakes of extra soy sauce.
TO FREEZE: Place leftovers in an airtight, freezer-safe storage container and freeze for up to 3 months. Let thaw overnight in the refrigerator, then reheat as directed above.