This recipe is from a cookbook, The Food Lab. It will require you to invest a fair amount of time in it, but will reward a very rich and balanced flavor.
Shrimp dry with paper towels and peel. Set the shells aside, don't throw them away.Remove the intestinal vein from the shrimp (ideally do this on both sides - the back and belly side, but if you are lazy, one is enough).
Crush 2 cloves of garlic with a knife and add to the shrimp shells.Cut 2 cloves into thin slices and set aside, and grate the remaining 2 cloves on a fine grater and add to the shrimp.
Add 2 tbsp of extra-virgin olive oil to the shrimp. Finely chop the parsley stalks and add to the same. Finely chop the parsley greens and set aside.
Fill a saucepan with water, salt with 2-3 large pinches of salt and put on the fire.
Heat the remaining olive oil in a frying pan over medium-low heat, add the shrimp shells and mashed garlic (right along with the skins), fry for 5 minutes, stirring occasionally. Make sure that the oil does not overheat and in no case begin to smoke. Sieve the oil, discard the shells and garlic.
At this point the water should already be boiling, add the dry pasta to it and stir for about a minute. Time it, and cook the paste a minute to a minute and a half less than what is indicated on the package.Pour the infused oil back into the skillet and heat over medium-low heat. Add garlic cut into thin slices, fry for about 30 seconds, stirring frequently. Add the shrimp and cook for another 40-50 seconds, flip the shrimp over. Cook for 30 seconds and then add butter and squeeze the juice of half a lemon into the pan. Cook for a minute stirring constantly. Turn off the heat.Grate the zest of half a lemon, salt, pepper, stir and set aside.
When ready, drain the pasta, saving about 100-120 ml in a separate container. Return the pasta to the pot.Place on medium heat. Add the shrimp along with the sauce, add half the water to the pasta and cook constantly stirring for about 2 minutes. If the pasta seems dry, you can add more water.Turn off the heat, add the parsley, stir again and serve immediately.