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1
Remove the sausage from its casings and break into small chunks. Heat the olive oil in a wide, heavy pan with high sides over medium-high heat. When it's smoking, add the sausage and saute, stirring occasionally, until nicely browned - about 5 minutes. Remove the sausage with a slotted spoon and keep warm on a plate.
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2
Reduce the heat to medium and add the garlic to the pan. Cook for a couple of minutes, until the garlic is softened and fragrant (do not burn it). Add the wine and stir to scrape up any brown bits on the bottom of the pan. Reduce the wine by about two thirds, then add the chicken broth and return the sauce to a boil. Reduce until the sauce is dark and somewhat syrupy, 7 to 10 minutes.
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3
While the sauce is reducing, bring a large pot of salted water to a boil over high heat. Fill a small saucepan with about 1/2 an inch of water, salt it generously and set it over high heat. When it begins to boil, add the broccoli and stir once. Cover the pot and cook the broccoli until just barely tender, a minute or two. Drain the broccoli and set aside.
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4
Add the thyme leaves to the sauce and then remove from the heat while you cook the pasta. Once the water is boiling rapidly, add the linguine and cook until al dente, according to the package directions. Reserve about a cup of the pasta water before draining the pasta. Add the drained pasta to the sauce and set over medium-low heat. Fold gently to distribute, and then add the butter and a little of the pasta water, tossing to coat. Add the reserved sausage, broccoli and Parmesan and toss again gently, adding enough pasta water to make the sauce silky but not watery. Taste for seasoning and add salt if necessary. Serve immediately, passing the Parmesan at the table.
