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 of the pasta cooking water, then drain the pasta and toss 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, half of the minced garlic (4 cloves), red pepper flakes, lemon zest, 1/4 tsp salt and 1/4 tsp pepper. Cook, stirring, about 3–4 minutes until shrimp are opaque and just cooked through. Transfer shrimp and any pan juices to a plate and set aside.
In the same skillet over medium heat, melt the remaining 2 tablespoons butter. Add the remaining minced garlic (2 cloves) and cook 30 seconds until fragrant. Sprinkle in the flour and stir continuously for 1 minute to cook the flour.
Slowly whisk in the chicken broth, heavy cream, and pesto until smooth. Increase heat to medium-high and bring the sauce to a simmer, then simmer 3–5 minutes until slightly thickened.
Reduce heat to low and stir in 1 tablespoon lemon juice, then add 1/2 cup grated Parmesan and stir until melted. Return the cooked shrimp and any collected juices to the skillet, then add the drained pasta and toss to coat. Add reserved pasta water or extra cream a little at a time if the sauce needs thinning.
Taste and adjust seasoning with additional salt, pepper, or red pepper flakes as desired. Serve garnished with extra grated Parmesan, chopped fresh parsley, and a squeeze of lemon juice if using.