Bring a large pot of generously salted water to a boil. Cook the linguine or spaghetti according to the package directions until al dente, reserve 1 cup of the pasta-cooking water, and then drain.

For fresh broccoli, add the florets to the pasta pot during the last 3–4 minutes of cooking. For frozen broccoli, add it during the last 2 minutes or steam it separately until just tender, then drain well.

Pat the shrimp dry with paper towels so they sauté instead of steam. Season them with the kosher salt, black pepper, and optional red-pepper flakes.

Heat 1 tablespoon of olive oil in a large skillet over medium-high heat. Arrange the shrimp in a single layer and cook for about 1–2 minutes per side, just until pink and opaque, then transfer them to a plate.

Reduce the skillet heat to medium and add the remaining 1 tablespoon olive oil along with the butter. Add the minced garlic and cook for about 30 seconds, stirring often and keeping it from browning.

Pour in the chicken broth or dry white wine and simmer for 2–3 minutes. Scrape up the browned bits from the bottom of the skillet as the liquid reduces.

Add the drained pasta and broccoli to the skillet, then toss with the lemon juice and Parmesan. Add the reserved pasta water a few tablespoons at a time until the sauce lightly coats the noodles.

Return the shrimp to the skillet and toss gently just until heated through. Finish with chopped parsley and additional Parmesan if desired, then serve immediately.