Remove beef tenderloin from fridge and let sit at room temperature for 30 minutes before cooking.
Place mushrooms in a food processor and season with salt and pepper; pulse to a rough paste.
Add a tablespoon of olive oil in a large skillet and heat it up over medium-high heat. I use my 12-inch cast iron skillet.
Transfer the mushroom paste to the skillet and cook for about 10 to 15 minutes, stirring frequently, until all liquid has evaporated.
Remove mushroom paste from the skillet and transfer to a plate to cool.
Meanwhile, wipe down the skillet and add remaining olive oil; set the skillet over high heat.
Season the beef with salt and pepper; when the oil is nearly smoking, sear the beef for about 1 to 2 minutes per side, or just until browned.
Remove beef from the pan and put aside to cool for 10 minutes.
Using a pastry brush, brush the beef all over with mustard. Set aside to continue to cool.
In the meantime, lay a sheet of plastic wrap on a work surface.
Shingle the prosciutto on the plastic wrap, slightly overlapping, into a rectangle that is big enough to cover the whole tenderloin.
Spread previously prepared mushroom mixture over the prosciutto.
Place the beef tenderloin at the bottom of the prosciutto slices.
Roll the prosciutto around the beef, keeping a tight hold on the edges of the plastic wrap. Form a tight barrel; then shape and twist the ends of the plastic wrap to secure it.
Put in the fridge and chill for 20 minutes.
Preheat oven to 425˚F.
Dust your working area with flour; roll out the puff pastry on your work surface and, using a rolling pin, shape it into a rectangle large enough to cover the beef tenderloin.
Remove beef from the fridge.
Remove the plastic wrap from around the beef, then lay the beef at the bottom of the rolled out pastry.
Brush the perimeter of the pastry with a bit of the beaten eggs.
Tightly roll the pastry around the beef.
Brush the seam with a bit of the egg yolks.
Transfer the beef to a foil-lined baking sheet, seam-side down, and brush the beef all over with the egg yolks.
Chill in the fridge for 15 minutes.
Remove from fridge and again brush all over with the remaining egg yolks.
Sprinkle with sea salt.
Bake for 40 to 45 minutes, or until pastry is golden and the beef’s internal temperature registers at 120˚F.
Remove from oven and let rest 15 minutes before cutting.
Cut and serve.