Make the filling: In a large skillet over medium heat, melt the butter. Add the fennel and season lightly with salt and pepper; cook, stirring frequently, until softened slightly, 5 to 6 minutes. Stir in the mustard and turn off the heat. Let cool slightly.
Set an overturned metal baking sheet into the center rack of the oven and preheat the oven to 400°F.
In a small bowl, beat the egg with 1 teaspoon water.
Line a large, flat cutting board with a piece of parchment paper. On a lightly floured work surface, roll out the puff pastry to a thin sheet about 14 by 15 inches in size. Carefully transfer to the parchment-lined board. Starting at one of the shorter sides, top one half of the pastry dough evenly with the mortadella slices, leaving a 1 inch border around the edges. Add half of the cheese, then the fennel mixture, then the remaining cheese. Brush the edges of dough nearest the filling with the egg wash. Then carefully fold the other half of the dough over the fillings, stretching lightly as needed to cover. Crimp with a fork to seal.
Brush the entire top and edges of the dough with the egg wash. Sprinkle the dough lightly with salt and pepper. Score the surface of the pie a few times with a paring knife to help steam escape.
Carefully slice the pie and the parchment paper onto the preheated pan. Bake until golden brown on top and bubbling on the inside, about 25 minutes.
Remove and let cool 5-10 minutes. Serve warm, cut into squares and with more Dijon mustard on the side.