Lightly grill or blanch asparagus. Cut tips to 4 inches in length. Chop the remaining asparagus and set aside for the filling.
Melt 2 teaspoons butter in skillet. Sauté leeks. Season with salt and pepper and a bit of fresh thyme. Add the chopped asparagus as the leeks soften and brown slightly. Continue for another minute. Set aside.
Melt 2 teaspoons butter in skillet. Sauté the chopped mushrooms. Add to the leeks and asparagus. Cool the mixture to room temperature.
Melt 2 teaspoons butter in skillet. Lightly sauté the whole mushrooms being used for garnish. Set aside.
Preheat oven to 375F.
Cut the phyllo sheets in half crosswise. Brush each layer with melted butter and a small dab of Dijon mustard. Lay them staggered over each other so they are not perfectly aligned and to create a ruffled edge. Place the prepared phyllo into a 14-inch by 4 3/4-inch rectangular flan pan with a removable bottom.
Sprinkle in half of the grated mozzarella and gruyere cheese. Dot with the goat cheese. Sprinkle with fresh thyme and the minced garlic.
Cover with the leek, mushroom and asparagus mixture, followed by the remaining mozzarella and gruyere cheese. Sprinkle with fresh thyme and pepper.
Garnish with the prepared asparagus tips and whole mushrooms. Drizzle lightly with olive oil.
Bake in the oven for 35 minutes until the phyllo is golden and the filling is baked. If the phyllo is golden, but the filling needs more time, cover the tart with foil until complete.