Use a medium sized saucepan to combine the milk, cream, garlic, three thyme sprigs, and 1 bay leaf. Place over medium-low heat. Heat until it almost, but not quite, reaches a boil. Remove from the heat, cover, and allow it to steep.
Add the olive oil to a skillet and place it over medium heat. Add the remaining thyme and the bay leaf. Add the mushrooms and leeks and sauté for about five minutes, until the liquid is evaporated. Season with salt and pepper.
Butter a large, oven-proof casserole dish. Arrange about a third of the potatoes on the bottom of the dish, so that they are overlapping. Season the potatoes with chopped parsley, chives, salt and pepper.
Top with half of the mushroom mixture. Place another third of the potatoes on top of the mushroom layer, then another layer consisting of the rest of the mushrooms. Then top with the last layer of potatoes on top.
You should have three layers of potatoes and two layers of mushrooms. Press the layers down with a spatula to condense.
Strain the bay leaf, garlic, and thyme out of the milk and pour the milk over the potatoes.
Sprinkle the surface with the Asiago cheese.
Cover with foil and bake in a preheated oven at 375 degrees for about one hour, or until the potatoes are fork tender.
Uncover and bake for an additional 15 minutes until the cheese is well melted and the gratin is golden around the edges.
Let stand ten minutes before cutting. Garnish with fresh parsley.