Pat the filet mignon dry with a paper towel and season with salt and pepper. Heat a large cast iron pan over medium-high heat, add oil and heat. Add the filets and sear until golden brown on all sides, about 2 minutes total.
Spread with mustard
Remove to a plate and set aside. Brush or spread the Dijon mustard over the filets and refrigerate until cool, about 15 minutes.
Return the pan to medium heat, add the butter and allow to melt. Add the shallots, mushrooms and thyme, stirring occasionally until the mushrooms reduce in size and are no longer watery, about 10 to 12 minutes.
Add the vermouth and stir to combine. Continue to cook until the mushrooms are dry. Remove from heat and discard the thyme. Transfer to a bowl and refrigerate until cool.
Stretch out 2 sheets of plastic wrap on a workspace. Lay out 2 slices of prosciutto on each so they are just overlapping. Spread the mushroom mixture over each. Top each with a piece of the filet mignon.
Chill the beef
Wrap the prosciutto around the beef and tightly wrap with the plastic wrap. Chill in the refrigerator for 1 hour and/or up to overnight. This ensures the meat will not overcook.
Wrap in puff pastry
Preheat the oven to 425ºF (220ºC). Cut the piece of puff pastry in half. Lightly dust a clean work surface and roll out the puff pastry. Unwrap the beef from the plastic wrap and place in the center of each piece of pastry.
Trim puff pastry
Fold the puff pastry over the beef, pinching to seal it on all sides, trimming any excess puff pastry from the bottom. Place it seam side down on a parchment paper-lined baking sheet.
Brush with egg wash
In a small bowl whisk together the egg and 1 tablespoon water. Brush the puff pastry with the egg wash and use a sharp knife to cut a few slits in the top of the pastry.
Bake mini beef Wellington
Bake until a thermometer inserted into the center reads 130°F (54ºC), about 25 to 30 minutes for medium rare. If needed, broil for a minute for color.
Let cool for 10 to 15 minutes at room temperature then cut in half and serve.