Fill a large pot with water and salt it generously until it tastes like the ocean. Bring to a rolling boil. Add pasta and cook according to package directions until al dente. Before draining, reserve 1/2 cup of pasta water. Drain pasta and set aside.
When pasta has 5 minutes left to cook, heat a large skillet over medium-high heat. Add butter and let it melt completely.
Sprinkle flour over melted butter and whisk constantly for 2 full minutes to create a roux. This cooks out the raw flour taste and creates the base for your sauce.
Add minced garlic to the roux and stir constantly for 30-60 seconds until fragrant but not browned. Watch carefully to prevent burning.
Pour in chicken broth and let it bubble for about 1 minute, stirring occasionally.
Gradually pour in heavy cream while whisking constantly. Let the sauce simmer gently for 3-4 minutes, stirring every 20-30 seconds, until it thickens enough to coat the back of a spoon.
Remove from heat and stir in freshly grated parmesan cheese until melted and smooth. Taste and season with salt and black pepper as desired.
Add drained pasta to the skillet and toss with tongs until every strand is coated with sauce. If sauce is too thick, add reserved pasta water 1 tablespoon at a time until desired consistency is reached.
Divide into bowls and garnish with fresh parsley and extra parmesan if desired. Serve immediately while hot.