Before cooking, make sure you take the fillet out of the fridge and let it come to room temperature. Season the fillet with salt and pepper.
In a large sauté pan, heat the olive oil over medium-high heat. Add the beef and sear it on all sides until it is nicely browned. If you want your beef wellington medium-rare, set it aside to rest now. If you would like it medium, place it in the oven for 10 minutes, then remove it and set aside to cool before you place it in the refrigerator. You can do this ahead of time.
In a large sauté pan, heat the extra virgin olive oil and the butter over medium-low heat. Add the mushrooms with the thyme and cook until the mushrooms are soft and all their water has evaporated, a good 8 to 10 minutes. Then add the Armagnac and continue to cook down, being very careful that the alcohol does not catch fire. Season with salt and pepper to taste and set aside to cool.
Lay out a piece of plastic wrap on your work surface and then place another piece across the first to form a cross. Then take the prosciutto slices and lay them out on top in two rows, making sure to lay the prosciutto in the same direction as the top layer of plastic wrap. Make sure the slices overlap so that there are no gaps and they are wide enough to enclose the fillet.
Take two-thirds of the cooled mushroom mixture and place it in the centre of the prosciutto along its entire length. Using the back of a tablespoon, push the mixture out to the sides. Place the chilled fillet in the middle of the mushroom mixture and take the remaining mushroom mixture and spoon it across the top of the fillet. Lift the top layer of plastic wrap and pull it up around the sides of the fillet, wrapping it over the mushrooms and prosciutto. Do the same going lengthwise with the bottom layer of plastic wrap. Tighten the plastic around the beef fillet so that it almost forms a sausage and then place in the fridge for 15 minutes or so.
No need to grease a baking sheet, just line it with parchment paper. Lay one sheet of puff pastry on your work surface. Cut off one third, roll it out slightly with a rolling pin so it’s big enough to accommodate the width and length your fillet, and place it on the prepared baking sheet. Take the fillet from the fridge and carefully remove the plastic wrap. Place it in the centre of the sheet of puff pastry. In a small bowl, beat the egg and milk together and gently brush some of the mixture along the edge of the puff pastry. Roll out the remaining pastry and carefully place it over the meat. Make sure it is big enough to hang down over the fillet. With a fork, press down the top layer of pastry onto the bottom layer, crimping them together to seal. Take a sharp knife and decoratively score the top of the pastry, without cutting through it. Then brush the remaining egg and milk mixture over the top of the beef Wellington. Refrigerate for a minimum of 1 hour.
Preheat the oven to 180°C
Bake in the middle of the oven for 35 to 40 minutes, until golden brown. Let it rest for 10 to 15 minutes, then cut it into half-inch-thick slices and serve with a vegetable of your choice.