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To make the crêpe batter: Put the eggs, milk, and salt in a blender and blend until smooth (or use an immersion blender). Add the flour and blend for 20 seconds, then drizzle in the butter and blend for 10 seconds more.
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Let the batter stand for 20 minutes. Note: The batter could also be made a day ahead, covered tightly with plastic, and refrigerated. Whisk well before using, as the flour will sink to the bottom of the bowl.
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To make the filling: In a large frying pan set over medium-high heat, melt the butter. Add the mushrooms and a generous pinch of salt and cook, stirring, until the mushrooms release their liquid; continue cooking until the liquid evaporates.
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Cook until the mushrooms are tender and lightly browned, about 6 to 7 minutes. Remove from the heat, stir in the thyme, and season to taste with additional salt and pepper. Set aside.
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To cook the crêpes: Heat an 8" crêpe pan or non-stick frying pan over medium-high heat and brush with clarified butter (or oil). Lift the pan from the burner and ladle 1/4 cup of batter into the center. Immediately tilt the pan in all directions so the batter spreads into a thin, even circle.
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Return the pan to the burner and cook until the edges of the crêpe look papery and brown, about 20 to 30 seconds. With your fingers or an offset spatula, flip the crêpe and cook on the second side until browned. Transfer to a plate.
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Repeat, greasing the pan with additional butter as necessary, until all the batter has been used. You should have about 16 crêpes, though often the first one you make may stick and the last one may be a bit small; the goal is 14 good crêpes.
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Preheat the oven to 350°F and lightly butter a 9" x 13" baking dish.
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To assemble the crêpes: Working with one crêpe at a time, spread a teaspoon of mustard on one side. Top with a slice of ham, followed by a spoonful of cooked mushrooms, arranging them in a line down the center. Tightly roll the crêpe into a cylinder and place in the prepared baking dish.
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Repeat the filling and rolling until all the crêpes have been used. Cover the baking dish with foil and place in the oven to heat through while you make the sauce (about 20 minutes).
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To make the sauce: In a medium saucepan, melt the butter, then whisk in the flour and cook until pale brown and toasty smelling, about 30 seconds. Whisk in the warm milk.
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Cook, stirring frequently, until the mixture bubbles and thickens. Add the Gruyère and whisk until melted and smooth. Season to taste with salt and pepper.
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Remove the crêpes from the oven, move the rack to 5" from the top, and turn on the broiler.
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Spoon the sauce over and around the crêpes in the baking dish and top with the grated Parmesan. Slide the baking dish under the broiler and broil until the sauce is browned and bubbling, 1 to 4 minutes — don't walk away!
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Serve crêpes hot. Store any leftovers covered in the refrigerator for up to 5 days; freeze for longer storage.