Gather and prep all ingredients. Finely mince the garlic, grate the Parmesan cheese fresh, and measure the Dijon mustard, thyme, salt, and pepper into a small bowl.
Place a heavy-bottomed saucepan over medium heat. Add the butter and let it melt completely. Once foaming, add the minced garlic and cook for 30 to 45 seconds until fragrant, stirring constantly. Do not let the garlic brown.
Sprinkle the flour evenly over the garlic butter and whisk vigorously for 1 to 2 minutes until a thick paste forms. Cook until the roux is slightly yellow and smells nutty, but do not let it brown.
Remove the pan from the heat for 15 seconds. Return to medium heat and add the milk in small increments (2-3 tablespoons at a time), whisking until completely smooth after each addition.
Once all the milk is incorporated, whisk constantly and cook for 3 to 4 minutes until the sauce thickens noticeably and coats the back of a spoon.
Reduce heat to low. Whisk in the Dijon mustard, thyme, salt, and pepper until fully combined.
With the heat on low, add the grated Parmesan and whisk gently for 30 to 45 seconds until the cheese is completely melted and smooth.
Taste and adjust seasoning as needed. If the sauce is too thick, whisk in an additional splash of milk. Serve immediately over chicken cordon bleu, garnished with fresh parsley if desired.