Boil the pasta: Bring a large pot of salted water to a boil. Cook the pasta until just shy of al dente. Reserve 1 cup of pasta water, then drain.
Season the salmon: Pat the salmon dry.
Season all sides with 1/2 teaspoon salt and 1/4 teaspoon black pepper.
Sear the salmon: Heat 1 tablespoon olive oil in a large skillet over medium-high heat. Add the salmon in a single layer and sear 1 to 2 minutes per side until just cooked through. Transfer to a plate and tent loosely with foil.
Do not overcook.
Sauté the aromatics: Reduce heat to medium. Add the remaining 1 tablespoon olive oil and the butter. Stir in the shallot and a pinch of salt.
Cook 2 to 3 minutes until softened. Add the garlic and red pepper flakes; cook 30 seconds until fragrant.
Build the sauce base: Pour in the broth and scrape the skillet to release browned bits. Simmer 1 to 2 minutes to reduce slightly.
Add cream and lemon: Stir in the cream, lemon zest, and 2 tablespoons lemon juice.
Simmer on medium-low for 2 to 3 minutes, stirring often, until lightly thickened. Taste and season with salt and pepper.
Wilt the spinach: Add the spinach by handfuls, stirring until just wilted, 1 to 2 minutes.
Cheese and consistency: Stir in the Parmesan until smooth. If the sauce seems too thick, add a splash of reserved pasta water until it coats the back of a spoon.
Combine with pasta: Add the drained pasta to the skillet and toss to coat.
Gently fold in the seared salmon. Add the remaining 1 tablespoon lemon juice if you like it brighter. Adjust salt and pepper.
Finish and serve: Sprinkle with chopped parsley or dill.
Serve right away with extra Parmesan and lemon wedges.