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 to 2 minutes. Transfer it to the bowl of cold water.

Heat 2 tablespoons 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 the pasta in the water. Reduce the heat under the skillet to medium low; add the remaining 2 tablespoons oil and the garlic. Season with salt and cook, flipping the pieces occasionally, until they start to brown, 3 to 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 1/2 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.