Bring a large pot of salted water to a boil. Add the 8 oz. fettuccine and cook according to package instructions until al dente, usually about 10-12 minutes. Drain, reserving 1/2 cup of pasta water, and set aside.
While the pasta cooks, heat 1 tablespoon olive oil and 1 tablespoon butter in a large skillet over medium-high heat. Add half of the minced garlic and the shrimp. Season with salt, pepper, and red pepper flakes. Cook shrimp for 2-3 minutes on each side until pink and opaque. Remove shrimp from the skillet and set aside.
In the same skillet, melt the remaining 2 tablespoons butter with 1 tablespoon olive oil. Add the remaining garlic and cook until fragrant, about 1 minute. Sprinkle the flour over the butter and garlic, stirring constantly to form a roux. Slowly whisk in the low sodium chicken broth, ensuring there are no lumps. Bring the sauce to a simmer and cook until it thickens, about 2-3 minutes.
Reduce heat to low and stir in the heavy cream, prepared pesto, lemon zest, and lemon juice. Mix well until creamy and smooth. Add the shrimp back into the skillet and toss to coat in the sauce.
Add the cooked fettuccine to the skillet, tossing gently to combine. If the sauce is too thick, add reserved pasta water a little at a time until desired consistency is reached. Stir in the freshly grated Parmesan cheese.
Sprinkle chopped fresh parsley and additional Parmesan cheese over the top before serving. Serve immediately with a side of crusty bread or a green salad.