Prepare the tofu: Press tofu for 15–20 minutes, then cut into cubes. Pat dry and toss with avocado oil, garlic powder, and salt. Sprinkle with arrowroot and toss until lightly coated.
Air-fry: Arrange in a single layer in a preheated air fryer at 400°F. Cook for 12–16 minutes, shaking halfway through, until golden and crisp. Work in batches if needed.
Mix the sauce: Whisk all sauce ingredients together until the arrowroot dissolves.
Brown the mushrooms: Heat the remaining teaspoon of avocado oil in a large skillet or wok over medium-high heat. Add mushrooms and cook for 4–5 minutes, stirring occasionally.
Cook the vegetables: Add garlic and ginger; stir for 30 seconds. Add the cabbage’s thicker white pieces and snow peas; cook for 2 minutes. Add the leafy cabbage and cook for another 1–2 minutes, keeping the vegetables slightly crisp.
Add sauce: Whisk the sauce again, pour into the skillet, and toss for 1–2 minutes until lightly thickened.
Finish: Gently fold in the tofu just before serving, or place it on top to preserve its crispness. Sprinkle with peanuts, green onions, and black sesame seeds. Add lime juice if desired.