How to Make It
In a large pot, season 4 quarts of water generously with 1/4 cup kosher salt. Bring water to boil. Cook pasta according to package instructions to al dente. Reserve 1 cup of the pasta water. Strain pasta, and set pasta and reserved pasta water aside.
In a skillet, melt 2 tablespoons of butter over medium high heat. Once butter has melted and begins to foam, add shrimp and cook, tossing occasionally, until the flesh is pink and the shrimp have just begun to curl, about 3 minutes. Remove from heat.
In a small bowl, whisk together milk and egg yolk until completely combined. Slowly stream in 1/2 cup of the reserved pasta water into the milk mixture, whisking to combine. Set aside.
In a large skillet over medium-high, melt remaining 4 tablespoons butter. Add garlic and stir into butter until foamy and fragrant, 1 minute. Add milk mixture and lemon peel to the skillet and constantly whisk until the butter is no longer separated and the mixture begins to bubble and foam. Slowly sprinkle 1 cup of the grated Parmigiano-Reggiano into the liquid as you whisk. Allow sauce to bubble and continue stirring gently until mixture thickens slightly, about 5 minutes. Remove strips of lemon peel and discard. Add cooked pasta to the sauce, and gently toss pasta to evenly coat. Once pasta is well coated, add remaining 1/4 cup grated Parmigiano-Reggiano to the pasta and continue tossing until melted and well distributed. (Note: if the sauce seems to thick, add a little of the remaining reserved pasta water to loosen the sauce.)
Remove pasta from heat, add cooked shrimp to pasta, and season with freshly ground black pepper, if desired. Garnish with additional Parmigiano-Reggiano and fresh parsley immediately before serving.