Bring a large pot of salted water to a boil. Add 8 ounces uncooked pasta and cook according to package directions until al dente. When the pasta has about 2 minutes left on its cook time, begin the sauce.
While the pasta finishes, melt 1 tablespoon butter in a large skillet over medium heat.
Sprinkle in 1 teaspoon flour and stir constantly to make a roux. Cook for about 1–2 minutes, stirring often, until the raw flour smell is gone.
Reduce heat to medium-low, add 2–3 cloves minced garlic and 1/4 cup chicken broth, and stir. Let the mixture simmer for about 1 minute.
Whisk in 1 cup heavy cream. Increase heat to medium and cook, stirring occasionally, until the sauce thickens slightly (about 2–3 minutes).
Remove the skillet from the heat (or reduce to very low) and stir in 1/3 cup freshly grated parmesan until melted and smooth. Season with salt and pepper to taste.
Before draining the pasta, reserve about 1/4 cup pasta cooking water. Drain the pasta and add it to the skillet with the sauce. Toss until the pasta is evenly coated, adding a little reserved pasta water a tablespoon at a time if you need to loosen the sauce.
Serve immediately, garnished with chopped fresh parsley if desired.