In a large skillet, heat olive oil over medium high heat. Add diced onions.
While onions cook, cut sausage in half (lengthwise) and then cut crosswise slices.
Add sausage to skillet for 3-5 minutes, allowing it to just start to brown.
Add cubed potatoes and broth to skillet. Stir to combine, then cover and let simmer for about 5-7 minutes (just until potatoes just start to get tender).
Stir in broccoli and simmer an additional 3-5 minutes.
Then stir in about half the shredded cheese. Put the remaining cheese on top and cover to melt for 2-3 minutes.