Peel and finely chop the onion and carrot. Thinly slice the celery. Warm ½ tbsp olive oil in a large pan over a medium heat for 1 min, then add the veg. Season with salt and pepper. Fry for 5 mins till softened, stirring occasionally.
While the veg are cooking, slice the skin of 4 of the sausages (see our tip for what to do with the rest) and squeeze out the sausagemeat. Add to the pan. Turn up the heat and fry for 5 mins, stirring to break up any lumps, till the sausagemeat is golden.
Add most of the thyme leaves to the pan, reserving a pinch for garnishing. Pour in the tin of tomatoes. Cover. Reduce the heat and simmer for 10-15 mins till the sauce thickens.
Fill a large pan with boiling water. Bring up to the boil. Add a pinch of salt and the pasta. Simmer for 3 mins.
While the pasta simmers, break the broccoli into small, equal-sized florets. Add to the pasta pan and continue cooking for another 5 mins till the pasta is al dente (just cooked but with a little bite). Drain the pasta and broccoli. Tip back into the pan.
Taste the tomato sauce and add more salt and pepper if it needs it. Stir the sauce into the pasta and broccoli. Mix well. Put on the lid. Leave for 2-3 mins so the pasta and broccoli soak up the flavours.
Grate the Parmesan. Serve the pasta in large bowls topped with the grated cheese and remaining thyme leaves.