Bring a large pot of salted water to a boil and cook the fettuccine according to package directions until al dente. Reserve 1/2 cup pasta cooking water, then drain and toss pasta with a little olive oil to prevent sticking; set aside.
In a large skillet over medium-high heat, melt 1 tablespoon butter with 1 tablespoon olive oil. Add the shrimp, 4 minced garlic cloves, red pepper flakes, lemon zest, 1/4 teaspoon salt and 1/4 teaspoon black pepper. Cook, stirring, about 4 minutes until the shrimp are opaque and just cooked through. Transfer shrimp and any pan juices to a plate and set aside.
Do not wipe out the skillet. Over medium heat, melt the remaining 2 tablespoons butter. Add the remaining 2 minced garlic cloves and sauté about 30 seconds until fragrant. Sprinkle in the flour and cook, stirring, for 1 minute.
Turn the heat to low and slowly whisk in the chicken broth, then add the heavy cream and pesto, stirring constantly until smooth.
Increase heat to medium-high and bring the sauce to a simmer. Cook, stirring occasionally, until slightly thickened, about 3–5 minutes.
Reduce heat to low and stir in 1 tablespoon lemon juice and 1/2 cup grated Parmesan until melted. Return the cooked shrimp (and any juices) to the skillet, then add the drained pasta and toss to coat in the sauce. Add reserved pasta water or more heavy cream as needed to reach the desired consistency. Taste and adjust seasoning with additional salt, pepper, or red pepper flakes.
Serve garnished with additional freshly grated Parmesan, chopped fresh parsley, and a squeeze of lemon juice if desired.