Step 1

Bring a large pot of salty water to boil and prepare a bowl of ice water. Drop the broccoli into boiling water and cook until tender but still slightly firm, 2-3 minutes. Transfer the broccoli with a slotted spoon to the ice bath to cool quickly. Stir until cool, then drain the water. Return the water to a boil and cook the pasta until al dente, reserving 1 cup of the pasta cooking water before draining.

Step 2

In the meantime, heat half the olive oil in a large skillet over medium heat and add the sausage. Break up into bite-sized pieces (but not until crumbly) and cook until golden brown and cooked through. Remove to a plate, leaving the fat in the pan. Add the garlic slices and remaining oil, if necessary depending on how much fat the sausages gave, and cook over low heat until golden. Add the broccoli and toss, then add the sausage, sun-dried tomatoes, and thyme. Increase the heat to medium and cook until heated through.