If making the creamy lime-chili mayo, combine 1/2 cup mayo, 1/2 teaspoon lime zest, 3 tablespoons lime juice, and 2 teaspoons chili-garlic sauce in a bowl; season lightly with salt and pepper, whisk until smooth, cover, and chill.
Prepare vegetables: dice mushrooms, thinly slice green onions and separate whites from greens, dice red bell pepper, and mince ginger and garlic.
Heat 1 tablespoon oil in a large cast-iron pan over medium-high heat until hot.
Add diced mushrooms, diced red pepper, and the white parts of the green onions to the pan and sauté, stirring occasionally, until mushrooms soften, about 3–4 minutes.
Push the vegetables to the edges of the pan and add the ground turkey to the center; break it up and cook until mostly cooked through, 3–5 minutes.
Add minced ginger and garlic, and continue to cook and break up the turkey until no pink remains, about 2–3 minutes more.
While the turkey cooks, whisk together 2 teaspoons cornstarch and 1 tablespoon soy sauce until smooth, then add 3 tablespoons chicken broth, 1/4 cup hoisin sauce, and 2 tablespoons chili-garlic sauce; whisk until combined and set aside.
Add the shredded cabbage and 2 cups matchstick or shredded carrots to the pan; stir for 1 minute to combine.
Pour the prepared sauce over the turkey and vegetables and cook, stirring, until the sauce is thickened and glossy and the vegetables are tender, about 2–3 minutes.
Remove from heat and stir in the green onion tops. Serve the turkey mixture over cooked rice, garnish with lime wedges and cilantro, and drizzle with the creamy sauce if using.
