Heat oil in a large (5- to 7-qt.) Dutch oven or other heavy pot over medium-high. Add beef and onion. Cook, stirring often, breaking up beef with a wooden spoon, until it begins to brown and onion softens, about 10 minutes.
Add broth, tomatoes, and salt. Bring to a boil over high. Reduce heat to medium-low. Let simmer until flavors meld, about 15 minutes.
Add gnocchi; cook, stirring often, until just tender, 3 to 5 minutes. Remove from heat.
Divide evenly among four bowls. Garnish with parsley and serve with grated cheese, if using.
Browning meat before adding it to a soup or stew deepens the taste.