Sprinkle salt into a 10 to 12 inch frying pan over medium-high heat. Crumble beef into pan and cook, stirring often, until meat begins to brown, 3 to 5 minutes. Reduce heat to medium, stir in onion, and continue to cook, stirring, until onion is soft but not brown, about 5 minutes. Spoon off and discard excess fat.
Add garlic, tomatoes (break them up with a spoon) and their liquid, broth, oregano, and pepper. Bring mixture to a boil then stir in pasta. Reduce heat, cover and simmer gently until pasta is tender to bite, 10 to 12 minutes.
Meanwhile, squeeze as much liquid as possible from spinach. Stir spinach into pasta mixture just until heated through. Serve with cheese to add to taste.