Bring a pot of salted water to a boil and cook pasta until just shy of al dente. Drain, reserving some pasta water.
While the water is boiling, melt the butter in a large skillet over medium heat. Add the salmon, lemon juice, half of the lemon zest, and half of the chives. Cook for a few minutes, stirring occasionally. Stir in the cream and cook over low heat (the cream should be at a simmer) for about 5 minutes.
Add the drained pasta to the cream mixture. Cook for a few minutes, tossing to coat. Taste for seasoning. If the pasta looks a little dry, add pasta water a tablespoon or two at a time until it is creamy enough.
Serve immediately, topped with the remaining chives and lemon zest.