Combine cheeses in a small bowl; set aside.
Cover bottom of prepared dish with potato slices, slightly overlapping them.
Sprinkle with half of the thyme and a pinch each of salt and pepper.
Sprinkle with 1/3 of the cheese and half of the mushroom mixture.
Repeat layering process, using all but 2 tablespoons cheese.
Arrange remaining potato slices, slightly overlapping, around the edge of the dish.
Stir together milk and cream in a bowl, and pour over top.
At this point, dish can be refrigerated, covered, up to 1 day; let stand at room temperature 30 minutes before baking.