In a medium saucepan, heat the butter over medium-low heat until melted. Add the garlic and cook, stirring constantly, for 1 full minute, until the garlic is fragrant and beginning to turn golden.
Whisk in the flour until smooth, about 1 minute. Gradually whisk in the milk, chicken broth, salt, and pepper. Bring to a gentle boil, stirring constantly. Lower the heat and simmer, stirring gently, until the mixture thickens, 2-3 minutes. Add Parmesan cheese and stir once more to incorporate.