Start by gathering all your ingredients on the countertop. Boil 4 tablespoons of water and add the saffron threads to it, allowing the flavors to infuse.
In a large pot, bring water to a boil. Add a generous pinch of salt and the dried linguine. Cook according to package instructions until al dente. Reserve about 1 cup of the pasta cooking water, then drain the pasta.
In a frying pan over medium heat, combine the saffron water with the double cream. Let it simmer gently, stirring occasionally. Add the grated parmesan, lemon zest, and a pinch of salt and pepper. Stir until the cheese melts and the sauce thickens, about 5 minutes.
Add the peeled cooked shrimp to the sauce and cook for an additional 3-4 minutes until heated through. Be careful not to overcook the shrimp, as it can become rubbery.
Add the drained linguine directly into the pan with the sauce. Toss to combine, ensuring that every strand of pasta is coated. If the sauce is too thick, add a splash of the reserved pasta water to achieve your desired consistency.
Remove the pan from heat and stir in the egg yolks quickly to enrich the sauce. Top with chopped chives for a fresh finish. Serve immediately, garnished with extra parmesan if desired.