Press the tofu for at least 10 minutes under a weighted plate, then pat very dry. Wet tofu will not crisp and will spit in the pan.
Whisk the peanut sauce: peanut butter, kecap manis, lime juice, and sambal in a small bowl. Add hot water a little at a time until it pours off the spoon in a thick ribbon. Taste and balance sweet, sour, and heat. Set aside.
Heat 1 tablespoon oil in a nonstick or well-seasoned pan over medium-high. Lay in the tofu slabs and fry undisturbed for 3 to 4 minutes a side until deep golden and crisp. Season with the white pepper. Move to a plate.
Cut the fried tofu into bite-size cubes. Lower the heat to medium and add the second tablespoon of oil.
Beat the eggs, fold the tofu cubes through them, then pour the mixture into the pan and spread it out. Let it set on the bottom for 2 minutes, then fold or flip in sections until just cooked through. You want a tender, slightly crisp-edged omelette, not a dry one.
Pile the bean sprouts and cucumber on a plate. Slide the tofu omelette on top, cut into squares, spoon over the peanut sauce, and finish with the chopped peanuts. Serve right away while the edges are still crisp.