Cover the noodles with very hot tap water in a large bowl and stir to separate. Let them soak for 35 to 40 minutes, until they are pliable and limp but not fully tender, then drain. While the noodles soak, spread the tofu over a paper towel–lined baking sheet and let it drain for 20 minutes.

Pulse the chiles, garlic, and shallots in a food processor until they form a smooth paste, about 20 pulses, scraping down the bowl as needed. In a separate small bowl, whisk the broth, fish sauce, sugar, and lime juice together until the sugar is dissolved.

Adjust an oven rack to the upper-middle position and heat the oven to 200°F. Pat the tofu thoroughly dry with paper towels, season with salt and pepper, and toss with the cornstarch in a medium bowl. Transfer the coated tofu to a strainer and shake gently to remove any excess starch.

Heat 3 tablespoons of oil in a 12-inch nonstick skillet over medium-high heat until just smoking. Fry the tofu, turning as needed, until it is crisp and browned on all sides, 8 to 10 minutes. Transfer the tofu to a paper towel–lined plate and keep it warm in the oven.

Wipe the skillet clean with paper towels, add 1 tablespoon of oil, and heat over high heat until just smoking. Sauté the snap peas and bell pepper, stirring often, until the vegetables are crisp-tender and starting to brown, 3 to 5 minutes, then transfer them to a bowl.

Heat the remaining 3 tablespoons of oil in the empty skillet over medium-high heat until shimmering. Cook the aromatic chile paste until the moisture evaporates and the color deepens, 3 to 5 minutes. Add the drained noodles and the broth mixture and cook, tossing gently for 5 to 10 minutes, until the sauce has thickened and the noodles are tender and well coated.

Stir in the cooked vegetables and basil and cook for about 1 minute, just until the basil wilts. Serve the noodles immediately, topped with the crispy tofu.