Slice onions thinly. Cut beef and broccoli into bite size. Season beef with salt and pepper. Then coat beef with all-purpose flour.
Cook fettuccine according to package directions. Add broccoli 2-3 minutes before the fettuccine is cooked.
While the fettuccine is cooking, make sauce. In a skillet, heat olive oil and cook onion over medium heat until tender. Add beef and cook until beef are browned. Stir in Dijon mustard and half of milk. Simmer until thickened stirring constantly. When it thickened, add the rest of milk, reduce heat to medium-low heat and simmer for 5-7 minutes. Stir in cheese. When the cheese starts to melt, turn off the heat.
When pasta and broccoli are cooked, drain in a colander and put into the skillet. Toss to coat.