Peel the shrimp, saving the shells and the heads. Place them in a sauce-pan with the butter and sauté until they just begin to brown. Deglaze with a cup of water and simmer for 10 – 15 minutes until most of the water has evaporated. Strain and save the shrimp butter. Discard the spent shells. This can be done well in advance.
Cooking and serving;
Put a generously salted pot of water on to boil to cook the spaghetti. When it comes to a boil, start the spaghetti and set a timer. In the meantime, chop the peeled shrimp into small pieces and mince the chives.
While the spaghetti is cooking, gently cook the shrimp in a large sauté pan in the shrimp butter until they are just barely cooked through. Grate some lemon zest into the mixture using a microplane, add some lemon juice and salt to taste. Adjust as needed to achieve balanced flavor. As soon as the spaghetti is ready, strain and transfer to the pan with the shrimp and the butter. Toss well. It’s important to add a few tablespoons of the starchy spaghetti cooking water to the pan which will thicken and emulsify with the butter to make a smooth sauce. You can add more butter or pasta water as needed to achieve this.
Divide the pasta among two bowls, cover with shrimp and sauce, sprinkle with minced chives and serve immediately.