Cook noodles according to package directions. Drain and rinse under cold water. Set aside. Heat 1 tablespoon of the oil in a large flat-bottomed wok or cast-iron skillet over high heat. Add bok choy whites and cook, stirring, for 1 minute. Add bok choy greens and cook until wilted, about 1 minute more. Transfer all the bok choy to a medium bowl.

Add the remaining 1 tablespoon oil, scallion whites, ginger and chili paste to the pan. Add the ground pork and cook, crumbling with a wooden spoon, until it's no longer pink, about 5-8 minutes. Add the chicken broth, soy sauce (or tamari) and sugar. Mix the 1 tablespoon of cornstarch with the 1 tablespoon of water in a small bowl and add to the pan. Bring to a boil and cook, stirring, until slightly thickened, about 1 minute. Stir in the rice noodles and bok choy and cook, stirring, until heated through, about 1 minute. Serve topped with scallion greens and a squeeze of lime juice.