Preheat oven to broil. Melt 1 tablespoon butter and brush over the top of the salmon fillets; season with salt and pepper and place on a baking sheet. Broil fillets 7 minutes.
Meanwhile, bring a large pot of salted water to a boil. Cook pasta according to package directions until al dente. Reserve 1 cup of the pasta water and drain the rest.
Melt the remaining 3 tablespoons butter in a small sauce pan. Once the butter starts to bubble and foam, add capers, garlic, lemon zest, and half the lemon juice. Sauté until the garlic is fragrant and starting to brown, about 1 minute.
In a large bowl, combine the pasta, reserved pasta water, and the remaining lemon juice; toss to combine.
Divide the pasta among two dinners bowls; add steamed broccoli and top with broiled salmon fillets. Drizzle salmon and pasta with caper sauce. Garnish with freshly grated parmesan and parsley leaves.