Preheat oven to 350F. Thinly slice potatoes using a mandoline slicer. If you don’t have a mandoline, just slice the potatoes with a knife, trying to keep them as uniform as possible. Transfer the cut potatoes to a bowl.
Melt butter in a small bowl and pour over the potatoes. Toss until all potatoes are covered. Layer the sliced potatoes into a lightly greased muffin tin, adding about 1/3 cup into each cavity.
Add mozzarella, parmesan, thyme, salt, and pepper to a bowl and mix. Top the potatoes with the cheese mixture.
Cover the muffin pan with tin foil and transfer to the oven. Bake for 20 minutes. Remove the tin foil and bake another 5-10 minutes, until cheese is browned and potatoes are cooked through.
While the potatoes cook, make the mushroom sauce. Add butter, mushrooms, and garlic to a pan over medium heat. Cook for 3-4 minutes until mushrooms are cooked through. Transfer the mushrooms to a bowl.
Return the now empty pan to the stove, still over medium heat. Add chicken broth, heavy cream and dijon mustard. Bring to a boil and let it cook for 3-4 minutes, stirring occasionally. The mixture should thicken and begin to look like a sauce. If it isn’t thickening, mix 1/2 tsp of cornstarch and 1 tbsp of water in a bowl and whisk to combine. Add the cornstarch slurry to the sauce and stir.
When the sauce has thickened slightly, add the mushrooms back in. Turn off the heat.
Remove the potatoes from the muffin tin and top with the mushroom sauce. Enjoy!