Preheat oven to 375 degrees F.
In a 12-inch skillet, heat 2 Tablespoons oil over medium heat. Add the leeks and mushrooms and season with ¼ teaspoon salt. Cook until vegetables are tender, about 10 minutes.
Add spinach to the skillet and cook until wilted, 2 to 3 minutes.
Grease a 9-inch deep dish pie plate with remaining tablespoon of oil. Break each matzo sheet in half. Add warm water to a large baking dish. Dip matzo pieces into water, letting excess water drip off, until matzo is soft and pliable but not falling apart, about 30 seconds. Line the inside of the prepared pie plate with matzo pieces and up the sides, slightly overlapping pieces as needed.
In a large bowl, beat the eggs and milk. Stir in the cooked vegetables and season with ½ teaspoon salt and pepper. Pour filling over matzo crust. Dot with goat cheese, if using. Bake for about 40 minutes, until the center is set and springy. (Push on the very center of the quiche and make sure your fingers don’t sink, but spring back.)