Heat 1 tablespoon of oil in a large non-stick pan or skillet over medium heat. Add garlic and mushrooms.
Sauté stirring occasionally until mushrooms are tender. Lightly season with salt and pepper and set aside in a bowl together with all the juices.
Cut a lengthways pocket to each chicken breast. Season the chicken breast with salt, pepper, and seasonings on both sides and into the pocket too.
Stuff the pockets with mozzarella slices and sautéed mushrooms, then seal with a toothpick. It's ok if you have any leftover mushrooms, we'll add them later on to the sauce.
Heat again the pan and add in the remaining oil. Add chicken and sear until golden on both sides, about 6-8 minutes in total. Set aside on a plate.
To the same skillet, add minced garlic and cook over medium heat for no longer than 1 minute, or until fragrant.
Add in the mustard, flour, and whisk for a bit. Pour the milk little by little, whisking after each addition to incorporate and form a thick sauce.
Once all the milk is added and there are no visible lumps, stir in the ground parmesan and season with a pinch of salt and pepper. If you have any remaining sautéed mushrooms and juices, now it's the time to add them to your sauce.
Return the chicken to the pan and into the sauce, and spoon the sauce over.
Cover the pan and cook over low heat for 12-15 minutes, or until the chicken is cooked through and the internal temp shows 165F.
Garnish with freshly chopped parsley and serve warm.