In a large pot of boiling salted water, cook spaghetti until al dente. Reserve 1 cup pasta water; drain pasta, and return to pot.
Meanwhile, heat 1 tablespoon butter in a large skillet over medium-low. Add leeks and garlic; season with salt and pepper. Cook, stirring occasionally, until leeks are tender, 6 to 8 minutes. Add leek mixture to pasta in pot (reserve skillet).
Heat 1 tablespoon butter in skillet over medium. Add scallops and lemon zest; cook, tossing, until scallops are cooked through, 2 to 3 minutes. Transfer to pot with pasta. Add tarragon, remaining 2 tablespoons butter, and lemon juice; toss, adding enough pasta water to create a thin sauce that coats pasta. Serve immediately.