For the crepes: Combine the eggs, milk, butter, salt and 1/2 cup water in the blender and blend for 7 to 10 seconds. Add the flour and blend until just combined, about 5 seconds, then stir in the herbs. Rest the batter in the refrigerator for 1 hour.
For the mushroom filling: Meanwhile, finely chop half of the shiitake and cremini mushrooms and set aside.
Heat a 12-inch saute pan over medium-low heat and add 2 tablespoons butter. When it has melted, add the onion and cook until translucent, about 5 minutes. Add all the mushrooms
(sliced and chopped), salt and pepper to the pan and cook, stirring occasionally, until the mushrooms are soft and the entire mixture has reduced to about one third of its original volume, 4 to 5 minutes.
Add the milk and cook until the mixture becomes a loose paste, about 2 minutes. Add the tarragon, chives and provolone and cook for another minute.
Heat the oven to 200 degrees F or its lowest setting. And set a cooling rack inside a half-sheet
Place an 8-inch nonstick pan over medium-low heat and coat with cooking spray or 1 tablespoon butter. Pour 1 ounce of the batter into the center of the pan and swirl the pan to spread it around evenly. Cook for 30 seconds, or until the edges of the crepe begins to turn up. Flip (you can use a small offset spatula) and cook for another 10 seconds.
Place the crepes in the oven on the cooling rack setup and cover with parchment until all the
crepes are ready.
Remove the crepes from the oven and turn broiler on high.
Spread a thin layer of the mushroom mixture on a crepe. Top with Parmesan and another crepe. Transfer to a broiler-proof plate. Repeat with 1 more layer, finishing with Parmesan. Build another crepe on another broiler-proof plate. Put the plates into the oven (if they can fit) and broil until the cheese is just browned, about 1 minute.