Cook the pasta according to the package directions; drain and return it to the pot.
Meanwhile, heat the butter in a large skillet over medium heat. Add the leeks, ½ teaspoon salt, and ¼ teaspoon pepper and cook, stirring occasionally, until the leeks have softened, 3 to 5 minutes. Add the shrimp and lemon zest and cook, tossing frequently, until the shrimp is opaque throughout, 4 to 5 minutes more.
Add the cream and ½ teaspoon salt to the pasta in the pot and cook over medium heat, stirring, until slightly thickened, 1 to 2 minutes. Add the shrimp mixture and the spinach and toss to combine.