Put the noodles in a large bowl and cover with very hot water. Let soak until just pliable, about 5 minutes. Transfer the noodles to a colander and drain, shaking and tossing the noodles once or twice.
Meanwhile, in a small bowl, whisk the fish sauce, brown sugar, lime juice and chiles.
In a large nonstick skillet, heat 3 tablespoons of the oil until shimmering. Add the shallots and garlic and cook over high heat, stirring occasionally, until lightly browned, about 3 minutes. Add the pad thai noodles and stir-fry until heated through, about 2 minutes. Add the shrimp and cook, stirring occasionally, until they begin to curl and turn pink, about 2 minutes. Scrape the noodles and shrimp to one side of the pan and add the remaining 2 tablespoons of oil to the empty side of the skillet. Add the eggs and cook, stirring occasionally, until nearly set, about 1 minute. Add the scallions and toss everything together, keeping the eggs relatively intact. Add the fish sauce mixture and stir-fry until the noodles are evenly coated, 2 to 3 minutes. Transfer the pad thai to a platter. Top with peanuts, cilantro and bean sprouts and serve with lime wedges.