In a saucepan, combine milk, cream, garlic, 3 thyme sprigs and 1 bay leaf over low-medium heat. Heat to just under a boil. Remove from heat, cover, and let steep while preparing the rest of the dish.
Coat a skillet with olive oil and place over medium heat. Add remaining thyme and bay leaf. Saute the mushrooms and leeks for 5 minutes to evaporate the liquid. Season with salt and pepper.
Butter a large casserole dish. Arrange 1/3 of the sliced potatoes on the bottom of the pan, overlapping them slightly. Season the potatoes with chopped parsley, chives, salt and pepper. Top potatoes with half of the mushroom mixture. Repeat layering 1/3 of the potatoes and remaining mushroom mixture. Arrange remaining potatoes on top. Press the layers down with a spatula to condense. Remove the solids from the infused milk and pour over the potatoes. Sprinkle the surface with cheese. Cover the pan with foil and place in a preheated 375 degrees F oven. Bake for 1 hour until the potatoes are tender. Uncover and bake for 15 additional minutes until cheese melts and gratin is golden around the edges. Let stand 10 minutes before cutting. Garnish with fresh parsley.