Cook the spaghetti according to the package directions.
Meanwhile, season salmon with ¾ teaspoon kosher salt and ¼ teaspoon black pepper. Cook in 2 tablespoons of the butter in a large nonstick skillet over medium-high heat until opaque throughout, 4 to 6 minutes per side; remove and flake. Add the leeks to the skillet and cook until tender, 4 to 6 minutes.
Toss the spaghetti with the salmon, leeks, remaining butter, and chives.
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