Pat the shrimp dry with paper towel.
Heat the olive oil and 2 tablespoons of butter in a large, deep skillet or Dutch oven over medium-high heat until the butter foams.
Add the shrimp in a single layer and cook for about 2 minutes, stirring once, until pink and almost opaque in the center.
Add the 2 minced garlic cloves, lemon zest, lemon juice, salt, and pepper and stir for 1 minute. Transfer the shrimp and their juices to a bowl, leaving the browned bits in the pan.
Pour in the chicken stock and scrape the bottom of the pan with a spatula to lift all the browned bits into the liquid.
Add the broken spaghetti, layering the halves in different directions.
Pour the water over the top, add the remaining minced garlic clove and the last tablespoon of butter, and press the pasta down until it is covered. Bring to a lively simmer.
Simmer for 10 to 12 minutes, stirring frequently and resting the lid on the pan between stirs, until the pasta is al dente and only a little loose liquid remains. Add a splash of hot water if the pan dries out.
Off the heat, stir in the Parmesan until it melts into a smooth sauce.
Return most of the shrimp with their juices, add the chili flakes, and stir through until the shrimp are hot and fully opaque.
Let the pasta stand for 2 minutes, then serve topped with the remaining shrimp, basil or parsley, and lemon wedges.