Put a large pot of salted water on to boil. FIll a large bowl with cold water. Blanch the broccoli in the boiling water until it softens to a tender crunch, 1-2 minutes (keep the water boiling). Transfer the broccoli with a slotted spoon to the bowl of cold water.
Heat 2 Tbs of the oil in a large skillet over medium-high heat for 1 minute. Add the sausage and saute, stirring, until it browns and is just cooked through, about 6 minutes. Transfer to a plate.
Put pasta in the boiling water. Reduce heat until the skillet to medium low; Add the remaining 2 Tbs oil and the garlic. Season with salt and cook, flipping the pieces occasionally, until they start to brown, 3-4 minutes. Raise the heat to high, drain the broccoli, and add it to the skillet, along with the sausage. Cook until everything is heated through, about 1 minute. Stir in the sun-dried tomatoes and thyme.
FInish cooking the pasta until it's just tender, about 10 minutes total. Reserve 1 cup of the pasta water. Drain the pasta and add it to the skillet with the broccoli. Raise the heat to high and toss well for 30 seconds. If the pasta appears dry, add ½ cup of the reserved pasta water. Stir in the lemon juice, Parmigiano, and scallions. Add more salt, pepper, or lemon juice to taste. Serve immediately, sprinkled with more Parmigiano.