In a large pot, bring 2 quarts water to a boil. Add the noodles and cook, stirring occasionally, until just shy of tender. Drain in a colander and rinse under cold water until cool to the touch. Drain again, shaking the colander to remove as much water as possible; set aside.

In a 12-inch nonstick skillet over medium-high, heat 3 tablespoons oil until barely smoking. Add the noodles in an even layer and cook, undisturbed, until they release easily from the pan and are crisped and browned on the bottom, about 5 minutes (check on browning after about 3 minutes by carefully lifting an edge with a spatula and looking underneath; reduce the heat if the noodles are coloring too quickly). Slide the noodles out of the pan onto a platter. Add 1 tablespoon of the remaining oil to the pan, then carefully flip the noodles back into the pan to cook the second side. Cook over medium until browned and crisped, about 6 minutes. Return the noodles to the platter and reserve the skillet.

In a small bowl, stir together the cornstarch and 2 tablespoons water; set aside. In a medium bowl, toss the shrimp with ¼ teaspoon each salt and pepper. In the same skillet over medium-high, heat 1 tablespoon of the remaining oil until shimmering. Add the shrimp and cook, stirring occasionally, until just pink on the exteriors but still translucent in the centers, 1 to 2 minutes. Return the shrimp to the bowl in which they were seasoned.

To the same skillet over medium, add the remaining 1 tablespoon oil, the ginger, garlic, scallion whites, mushrooms and carrot. Cook, stirring, until the mixture is fragrant, about 1 minute. Add the broth and fish sauce; bring to a simmer. Stir the cornstarch mixture to recombine, then stir it into the vegetable-broth mixture. Return to a simmer and cook, stirring, until the sauce is lightly thickened, about 1 minute.

Add the bok choy and shrimp along with any accumulated juices; cook, stirring, until the bok choy is tender-crisp and the shrimp are opaque throughout, about 2 minutes. Taste and season with salt and pepper. Scrape the mixture onto the noodles and sprinkle with the scallion greens.