Preheat the oven to 350°F and line a rimmed baking sheet with parchment paper.
Season the filets with salt and pepper. In the skillet, heat the oil until shimmering. Sear the steaks over high heat until you get a dark brown crust, about 3 minutes on each side. It should still be rare within—the steaks will continue to cook in the oven. Let cool to room temperature.
To assemble the Wellington, cut the puff pastry sheets in half. Using a pastry brush, brush the edges of each piece with the beaten eggs.
Working one at a time, smear 1 tablespoon of mustard all over a filet. Mold ½ cup of the mushroom duxelles around the filet (like molding Play-Doh).
Wrap the filet in 2 slices of prosciutto and place in the center of a piece of puff pastry.
Gently place a second piece of puff pastry on top of the filet—egg-washed edges to egg-washed edges. Gently press out any air bubbles before sealing the edges. Trim any excess to make a perfect square (save scraps to decorate the top of the Wellington—adhere with more egg wash). Crimp the edges with a fork and brush the pastry all over with the egg wash. Repeat with the remaining puff pastry, filets, mustard, duxelles, and prosciutto.
Transfer the Beef Wellington to the prepared baking sheet and bake for 20 minutes, or until the puff pastry is puffed and golden brown.
Remove from the oven and brush with the melted butter. Transfer to plates and serve right away.