Cook the soba noodles according to the package directions. Drain and transfer to a large zip-top plastic bag. Add 2 teaspoons of the oil, seal the bag, and shake to coat the noodles.
Meanwhile, heat the remaining 2 teaspoons oil in a large skillet over medium heat. Add the ginger and cook, stirring, for 1 minute. Increase the heat to medium-high, add the pork, and cook, stirring, until browned on all sides, about 5 minutes total. Add the salt and pepper and stir to coat the pork. Add the scallions and sesame seeds and cook, stirring, for 1 minute. Add the broth and soy sauce, reduce heat to medium, and simmer until the pork is cooked through, about 5 minutes. Remove from the heat and stir in the water chestnuts.
If you’re stopping here: Let cool to room temperature, then transfer the mixture to a large zip-top plastic bag or plastic container and refrigerate along with the bag of noodles for up to 3 days.
When you’re ready to eat: Combine the pork mixture and soba noodles in a large skillet set over medium heat and simmer for 5 minutes to heat through. You can also reheat the noodles and pork mixture together in the microwave for 3 to 5 minutes on HIGH. Meanwhile, chop the cilantro. Remove from the heat, stir in the cilantro, and serve.