Line a plate or cutting board with paper towels. Remove the tofu from its package, wrap in the towels, and press under a heavy skillet or tofu press for 15 minutes to drain excess moisture.
While the tofu presses, preheat the oven to 425°F (220°C). Arrange a wire rack on a rimmed baking sheet and lightly brush or spray the rack with oil so the tofu won't stick.
Unwrap the pressed tofu and cut into 3/4-inch cubes. In a large bowl, toss the cubes with 1 tablespoon oil, cornstarch, salt, and pepper until they are evenly coated in a thin, dry layer.
Spread the coated tofu in a single layer on the prepared rack. Bake for 12 minutes, flip or shake the rack to turn the pieces, then bake another 12 minutes. For extra crunch, switch the oven to broil and broil 1–2 minutes more, watching closely so they don't burn.
While the tofu bakes, whisk together the peanut butter, soy sauce, rice vinegar, maple syrup, sesame oil, garlic, ginger, and red pepper flakes. Add warm water a tablespoon at a time until the sauce reaches a pourable but thick consistency.
When the tofu is crisp and golden, remove it from the oven. Either toss the warm tofu gently with about half the peanut sauce to coat (this softens the crust slightly) or serve the sauce on the side for dipping to keep the cubes ultra-crispy.
Plate the tofu over rice, noodles, or a bed of greens. Spoon extra sauce over top or serve on the side. Sprinkle with chopped peanuts, sliced green onions, and cilantro. Serve with lime wedges to brighten the flavors.