To prepare salad: Halve tofu crosswise. Fold a clean kitchen towel in half and place it on a cutting board. Set the tofu on the towel. Put another folded towel and a weight (such as a heavy skillet) on the tofu; let drain for 15 minutes.

Meanwhile, place potatoes in a large saucepan and fill with water. Cover and bring to a boil over high heat. Reduce heat to a simmer and cook until the potatoes are tender when pierced with a fork, 5 to 8 minutes. Transfer the potatoes to a colander. Return the water to a boil and cook noodles according to package directions, stirring often, until just tender. Drain well and rinse under cold water. Transfer to a bowl and drizzle with 2 tablespoons oil; toss to coat.

Cut the pressed tofu into 3/4-inch cubes. Heat 1 tablespoon oil in a large cast-iron or nonstick skillet over medium-high heat until shimmering. Add the tofu in a single layer. Cook, without stirring, until starting to brown on the bottom, 2 to 3 minutes. Stir and continue cooking, stirring frequently, until brown on all sides, 6 to 8 minutes more. Transfer to the bowl with the noodles.

Soak scallions in cold water in a small bowl for 5 minutes. Drain and add to the noodles along with the potatoes, cabbage, cilantro and crushed red pepper; mix well with your hands.

To prepare dressing: Soak tamarind in hot water for 5 minutes, using a spoon to break up the pulp. Pour through a fine-mesh strainer into a bowl, pushing on the solids and scraping the bottom of the strainer to extract as much tamarind as possible. Whisk oil, garlic, ginger, lime juice, salt, sugar and turmeric into the tamarind water. Pour the dressing over the salad and toss to coat. Top with shallots and sprinkle with chickpea flour.