Cook linguine according to package directions until al dente. Reserve about 1 cup of pasta water before draining.
While the pasta cooks, melt butter in a large skillet over medium heat. Add minced garlic and red pepper flakes (if using) and cook until fragrant, about 1 minute.
Add the shrimp to the skillet and season with salt and pepper. Cook for 2-3 minutes per side, until pink and opaque. Remove shrimp from skillet and set aside.
Pour chicken broth and heavy cream into the skillet. Bring to a simmer and cook for 2-3 minutes, until slightly thickened.
Stir in lemon juice and Parmesan cheese. Cook for another minute until the cheese is melted and the sauce is smooth.
Add the cooked linguine and shrimp back to the skillet. Toss to coat with the sauce. If the sauce is too thick, add a little reserved pasta water until desired consistency is reached.
Stir in chopped parsley. Taste and adjust seasoning with salt and pepper as needed.
Serve immediately.
Notes
For an extra touch of flavor, you can add a splash of white wine to the sauce before adding the broth and cream.