Cook pasta in salted water until al dente. Reserve ½ cup pasta water before draining. Set aside.
Melt 1 tablespoon butter with 1 tablespoon olive oil in a large skillet over medium-high heat. Add shrimp, 4 minced garlic cloves, 1/2 teaspoon red pepper flakes, lemon zest, 1/4 teaspoon salt and 1/4 teaspoon pepper. Cook and stir for approximately 4 minutes or just until shrimp are opaque and cooked through. Scrape shrimp and garlic/juices to a plate. Once cool enough to handle, you can chop off shrimp tails if desired. Set aside.
To the same skillet (don’t wipe out), melt 2 tablespoons butter over medium heat. Add remaining 2 minced garlic cloves and saute 30 seconds. Sprinkle in flour and cook while stirring for 1 minute, then turn heat to low. Slowly whisk in chicken broth, heavy cream and pesto, stirring constantly until smooth.
Turn heat to medium high and bring the sauce to a simmer. Simmer until slightly thickened, stirring occasionally, approximately 3-5 minutes.
Reduce heat to low and stir in lemon juice followed by Parmesan cheese until melted. Stir in shrimp (and juices) and pasta until well coated in sauce, adding additional reserved pasta water or heavy cream if needed to reach desired consistency. Taste and season with additional salt, pepper and/or red pepper flakes to taste.
Garnish with freshly grated Parmesan Cheese, fresh parsley and lemon juice (optional).