Bring a large pot of salted water to a boil and add the linguine, cooking until al dente according to package directions (usually 9-11 minutes). While the pasta cooks, mince the garlic cloves finely, grate the parmesan cheese, chop the parsley, and measure out the heavy cream and reserved pasta water bowl. Before draining the pasta, reserve 1 cup of the starchy cooking water in a separate bowl—this will be crucial for adjusting the sauce consistency later. Drain the pasta and set aside.
Heat 1.5 tablespoons of olive oil with 1.5 tablespoons of unsalted butter in a large skillet over medium-high heat until the butter foams. Once foaming, add the shrimp and cook undisturbed for about 1 minute to develop a light golden color on one side. Season with 3/4 teaspoon salt, 1/2 teaspoon pepper, and 1.5 tablespoons Old Bay seasoning, then toss and continue cooking for another 1-2 minutes until the shrimp are just pink and cooked through (they should still be slightly firm—don't overcook or they'll toughen). Transfer the cooked shrimp to a clean plate.
In the same skillet with the shrimp drippings, add 2 tablespoons of fresh unsalted butter and let it melt over medium heat. Add the minced garlic and cook for just 30 seconds, stirring constantly—you want it fragrant but not browned, as burnt garlic becomes bitter. Pour in the heavy cream and bring to a gentle simmer, then add the freshly grated parmesan cheese, stirring constantly until the cheese melts completely and the sauce becomes smooth and velvety. I like to add the reserved pasta water gradually (about 1/4 cup at first) to reach the right sauce consistency—it should coat the back of a spoon but still flow slightly.
Add the cooked linguine from Step 1 directly to the cream sauce and toss gently to coat every strand, adding a splash more reserved pasta water if needed to achieve a silky, flowing consistency. Return the cooked shrimp from Step 2 to the skillet and fold them in carefully to distribute evenly throughout the dish. Taste and adjust seasoning with salt and pepper as needed. Remove from heat and finish with the fresh parsley, lemon zest, and chopped chives, tossing everything together one final time for a bright, fresh burst of flavor.