Cook the pasta in salted boiling water until it is just shy of "al dente". Drain and set aside. Prepare the sauce while the pasta is cooking.
Toss the broccoli with one teaspoon of the olive oil. and place in a grill basket. Cook 8 - 10 minutes until it is just tender crisp and lightly charred.
Cook the sausages on the grill whole until they are nicely browned. This takes about 10 minutes. Bring them in and allow them to cool slightly. Slice into 1/2 inch thick coins. Set aside.
While the sausage and broccoli cook, heat a large fry pan over medium high heat. Add the remaining olive oil to the pan and swirl to coat the bottom. Add the onions and cook until they begin to wilt and become translucent. Add the garlic and saute another minute or two.
Add the chicken broth and the thyme leaves to the pan. Cook for 3 minutes. Add the lemon zest and juice and allow to cook 2 more minutes.
Add the sausage and broccoli to the pan along with the half & half. Simmer for a few minutes.
Add the Romano Cheese and salt and pepper to taste. Cook until the cheese melts into the sauce. Serve in pasta bowls with a sprinkling of the finely grated cheese.