Make the filling: In a skillet over high heat, melt butter and sauté mushrooms until golden brown, about 8 to 10 minutes. Turn heat down to medium, add shallots, and cook 1 minute. Sprinkle flour evenly over vegetables and stir until coated. Add broth, stirring mixture until thickened. Add spinach and cook, stirring, until wilted, about 2 minutes. Remove from heat and stir in cheese. Season with salt and pepper.

Make the crêpes: Mist a crêpe pan with no-stick spray; warm over medium heat. For each crêpe, pour in ¼ cup batter, instantly swirling around pan to spread evenly to the edges in a thin film. Cook quickly until golden brown, 1 to 2 minutes, then turn and brown other side, about 1 more minute. Pile cooked crêpes on top of one another, keeping warm under an inverted plate or pot lid (if you must keep them warm a while, store in a low oven). Divide filling among crêpes, then fold around filling. Sprinkle with cheese.