Bring a large pot of water to a boil and stir in the kosher salt.
Chop broccoli into bite-size florets; peel and slice stems thin if using.
Boil pasta until almost al dente. Add broccoli for the last 2 minutes of cooking.
Reserve 3/4 cup pasta water, then drain pasta and broccoli.
Warm olive oil and butter in a large skillet over medium heat. Soften garlic for 30 seconds.
Pour in heavy cream and a splash of the hot pasta water. Simmer 2–3 minutes to slightly thicken.
Stir in pesto, lemon juice, red pepper flakes, black pepper, and sea salt. Reduce heat to low.
Fold in drained pasta and broccoli. Toss, adding Parmesan and more pasta water as needed until glossy and creamy.
Finish with lemon zest. Plate and sprinkle with pine nuts and extra Parmesan.