Start by bringing a large pot of salted water to a boil. Add the linguine and cook according to the package instructions until al dente, usually around 8-10 minutes. Drain the pasta, reserving about 1/2 cup of the pasta water for later.
In a large skillet over medium heat, melt the butter. Add the minced garlic and sauté for about 1 minute until fragrant. Be careful not to let it brown as it can become bitter.
Sprinkle the flour over the garlic and stir well to form a paste. Cook for about 1 minute, stirring constantly.
Gradually pour in the chicken broth while stirring to combine. Let it simmer for 2-3 minutes until slightly thickened.
Slowly add the heavy cream to the skillet, stirring continuously. Allow the mixture to come to a gentle simmer.
Stir in the freshly grated parmesan cheese, allowing it to melt into the sauce. If the sauce is too thick, add a splash of the reserved pasta water to reach your desired consistency.
Add the drained linguine directly into the skillet, tossing it with the creamy garlic sauce until well coated. Season with salt and pepper to taste.
Transfer the creamy pasta to serving plates or bowls. If desired, sprinkle with chopped fresh parsley for a pop of color and added freshness.