Bring a large pot of salted water to a boil and cook fettuccine according to package directions until al dente. Reserve 1/2 cup pasta water, drain, toss with a little olive oil, and set aside.
Heat 1 tablespoon butter and 1 tablespoon olive oil in a large skillet over medium-high. Add shrimp, 4 minced garlic cloves, red pepper flakes, lemon zest, salt and pepper; cook, stirring, until shrimp are opaque, about 3–4 minutes. Transfer shrimp and juices to a plate and set aside.
In the same skillet, melt the remaining 2 tablespoons butter over medium heat. Add the remaining 2 minced garlic cloves and cook 30 seconds until fragrant.
Sprinkle in the flour and cook, stirring, for 1 minute. Reduce heat to low and slowly whisk in chicken broth, then heavy cream and pesto until smooth.
Increase heat to medium-high and simmer the sauce, stirring occasionally, until slightly thickened, about 3–5 minutes.
Reduce heat to low, stir in lemon juice and 1/2 cup grated Parmesan until melted. Add the cooked shrimp with any accumulated juices and the drained pasta; toss to coat, adding reserved pasta water or a splash more cream to reach desired consistency. Taste and adjust seasoning with salt, pepper, and red pepper flakes.
Serve topped with extra Parmesan and chopped fresh parsley and an optional squeeze of lemon.
