This traditional Korean noodle dish is called japchae, and it's totally appealing to kids, given its sweetness, saltiness, and steakiness. Sweet potato noodles are transparent and wildly chewy; look for them in an Asian supermarket or in the international foods section of a large supermarket, or else substitute the closest approximation you can find.
Begin by reconstituting the mushrooms: in a small bowl, cover them with very hot tap water and set them aside.
Now cook the noodles in a large pot of boiling water for 4-5 minutes (it may be less or more if you're using a different type of noodle: you don't want to cook them beyond chewiness), then use tongs to pull them out into a colander and rinse them very well under cold running water. They will be very, very chewy, which is what you want. If you like, use scissors to cut them up a bit for easier eating.
In that same pot of boiling water, blanch the spinach: it will turn bright green almost immediately, at which point you should drain it in a colander and rinse it under cold water. Gather it into a ball and squeeze all the moisture out of it and, if you used large leaves of spinach, cut the ball in half with a knife. In a small bowl, dress the spinach with half the garlic, a teaspoon of the sesame oil, and the salt and set it aside.
Heat the oil in a large skillet over medium-high heat and add the steak, a spoonful of the soy sauce, and the rest of the garlic. Stir-fry until the meat is cooked, 3-4 minutes, then add the onion, the carrot, and the mushrooms, which you've drained and cut into slices. Cook until the onion is translucent, another 3 or so minutes, then add the scallions and stir-fry for another minute.
In a large bowl, combine the noodles, the spinach, the beef mixture, the remaining soy sauce, the remaining sesame oil, and the sugar. Mix it will -- you will likely need to use your hands for this -- then serve warm, sprinkled with the sesame seeds.