Slice tofu into 1/4- to 1/2-inch thick slices. Lay them on a clean towel covered with 2 layers of paper towels. Place a baking sheet on top and place a heavy weight on the baking sheet. Let sit for at least 30 minutes.
Remove lettuce core and soak, bottom down, in a bowl of cold water, about 1 hour. Remove from water and allow to drain. Remove the leaves carefully, working from the outside in. Use scissors to cut leaves into more shapely "bowls," if you like.
Preheat the oven to 375 degrees F (190 degrees C). Spray a baking sheet with cooking spray.
Mix soy sauce, vinegar, peanut butter, honey, water, and chile-garlic sauce together in a microwave-safe bowl. Heat in the microwave until warmed, 20 to 30 seconds. Stir peanut sauce until smooth.
Dip tofu slices in peanut sauce and place on the prepared baking sheet. Reserve remaining sauce. Spray tofu with cooking spray.
Bake in the preheated oven until dry, 45 to 60 minutes, flipping once or twice. Cool until safe to handle and chop into cubes.
Heat oil in a wok over medium heat. Add onion and cook until slightly softened, about 3 minutes. Add mushrooms; cook until onion is translucent, about 3 minutes more. Add garlic and ginger; cook until fragrant, about 1 minute more. Add tofu cubes and reserved sauce. Cook until heated through, about 3 minutes more. Add water chestnuts and green onions. Remove from heat.