Combine the broth and 2 ½ cups water in a large deep-sided skillet and bring to a rapid boil over high heat. Stir in the pasta and boil uncovered for 3 minutes less than the minimum on package.
Stir frequently with wooden spoon to cook the pasta evenly. Try to separate any pasta that sticks together — it's okay if a few noodles here and there stick, but you don't want a big ball bobbing around. And don't worry if some of the pasta sticks out of the water near the end of the cooking process — a lot of the stock and water will have evaporated at this point.
Reduce the heat to medium-high. Stir in the ham and cheese. Add a dash of salt, plus as much pepper as you like.
If there's a ton of sauce at this point, turn the heat down a notch and blend the cornstarch with 1 tablespoon of water in a separate cup and pour it into the sauce, cranking up the heat to medium-high again and stirring until the slurry is incorporated.
Stir in the spinach. Continue cooking uncovered, stirring frequently, until the pasta is al dente (it still has a bit of bite) and the sauce is creamy and thick, about 1 to 3 minutes. If the sauce thickens too much before the noodles are tender, turn off the pot, cover it, and let the steam build-up cook the noodles for a few minutes.
Transfer to large, shallow bowls and serve with a few extra grinds of fresh pepper.