Bring a large pot of water to a boil. Add 1 tablespoon of salt and cook the pasta according to package directions until al dente. Reserve 1 cup of pasta water before draining.

Pat salmon fillets dry with paper towels. Season both sides with salt and pepper. Heat olive oil in a large skillet over medium-high heat. Add salmon and cook for 4-5 minutes per side, until golden and cooked through. Remove to a plate and cover loosely with foil.

In the same skillet, reduce heat to medium. Add 2 tablespoons of butter. Add minced garlic and cook for 30 seconds until fragrant. Pour in white wine to deglaze, scraping up any browned bits. Let it simmer for 2 minutes. Stir in heavy cream, broth, lemon juice, and lemon zest. Bring to a gentle simmer.

Let the sauce simmer for 3-4 minutes to thicken slightly. Stir in capers, parsley, and Parmesan cheese until melted. Add the remaining 1 tablespoon of butter for extra richness. Season with salt and pepper to taste.

Add the drained pasta to the skillet with the sauce. Toss to coat thoroughly, adding reserved pasta water a little at a time if the sauce is too thick. Flake the cooked salmon into large chunks and gently fold into the pasta. Serve immediately.