Make the pastry by whisking together the gluten free flour, xanthan gum, and salt in a large bowl. Cut in the cold butter using a pastry cutter or your fingertips until the mixture resembles coarse crumbs. Add the egg and ice water and stir until the dough comes together.
Shape the dough into a flat rectangle, wrap it tightly in plastic wrap, and refrigerate for at least 1 hour. Season the beef tenderloin generously with salt and pepper. Heat the olive oil in a large oven-safe skillet over high heat and sear the tenderloin on all sides until deeply browned, about 1 to 2 minutes per side. Brush the seared beef all over with dijon mustard and set aside to cool.
In the same skillet, melt the butter over medium-high heat. Add the mushrooms and shallots and cook, stirring occasionally, for 6 to 8 minutes until all the moisture has evaporated and the mixture looks dry. Stir in the garlic and thyme, season with salt and pepper, and cook for 1 more minute. Spread the mixture onto a plate to cool completely.
Lay a large sheet of plastic wrap on the counter. Arrange the prosciutto slices in an overlapping layer on top of the plastic wrap. Spread the cooled mushroom mixture evenly over the prosciutto. Place the seared beef tenderloin at one edge and use the plastic wrap to tightly roll everything around the beef. Twist the ends to secure and refrigerate for 20 minutes.
Preheat the oven to 425°F and line a baking sheet with parchment paper. Roll out the chilled pastry dough on a well-floured surface into a rectangle large enough to wrap around the beef.
Remove the plastic wrap from the beef roll. Place the beef in the center of the pastry and fold the pastry around it, sealing the seam and the ends with the beaten egg. Place seam-side down on the prepared baking sheet and brush the entire surface with beaten egg.
Use a fork to poke a few small steam vents in the top of the pastry. Bake for 30 to 35 minutes, until the pastry is deep golden brown and the internal temperature reads 125°F for medium-rare. Let rest for 10 minutes before slicing and serving.