Heat the oven to 450°. For the filling, melt the butter in a large nonstick skillet over medium heat. Add both kinds of mushrooms and sauté them until they release their liquid and begin to brown, about 8 minutes. Add the shallots, sautéing them until they are softened, about 2 minutes, then add the garlic and the thyme and sauté the filling for 30 seconds. Season it with the salt and pepper. Stir in the wine and cook for about 2 minutes. Remove the skillet from the heat, stir in the ham, and set it aside.
To make the sauce, melt the butter in a heavy saucepan over medium-low heat. Add the flour and whisk for 2 minutes, then gradually add the milk, whisking to combine. Raise the heat to medium and bring the sauce to a boil. Let it simmer, still whisking, until it is slightly thickened, about 1 minute. Remove the sauce from the heat and stir in the salt, pepper, and nutmeg.
Stir 1/4 cup of the sauce into the filling. Arrange the tortillas on a work surface and spoon a quarter of the filling over half of each tortilla. Fold the tortilla in half, then fold it in half again to make a triangle. Stir 2 tablespoons of the cheese into the remaining sauce.
Coat a shallow 8- or 9-inch square baking dish with cooking spray and arrange the crepes in a single layer. Spoon the remaining sauce over the crepes. Sprinkle the remaining 1/4 cup of cheese on top.
Bake the crepes on the top rack until they are bubbling and golden, about 10 to 12 minutes. Let them rest for 5 minutes before serving.