Pat the tenderloin dry and season it all over with salt and pepper. Heat the olive oil in a large skillet over high heat.
Sear the beef on all sides, about 1 minute per side, until browned. Brush with Dijon mustard and transfer to a plate to cool.
Cook the Duxelles:
Melt the butter in the skillet over medium heat. Add the shallot and cook for 2 minutes.
Add the mushrooms, garlic, thyme, and remaining salt. Cook, stirring often, until the mushrooms release and then lose all their moisture.
Add the wine and cook until the pan is dry. Spread the duxelles on a plate and cool completely.
Wrap and Bake:
Lay plastic wrap on the counter and arrange the prosciutto in a slightly overlapping rectangle. Spread the cooled duxelles over it.
Place the beef in the center and use the plastic wrap to roll the prosciutto tightly around it. Chill for 15 minutes.
Roll the puff pastry into a rectangle large enough to enclose the beef. Unwrap the beef, place it on the pastry, and seal the edges. Chill for 30 minutes.
Heat the oven to 425 F (220 C). Brush the pastry with egg wash, score lightly, and bake for 35 to 45 minutes, until golden and the center reaches the desired temperature.
Rest for 15 minutes before slicing thickly and serving.
Notes
The mushrooms must be cooked dry or their moisture will soften the pastry.
Chilling the wrapped beef helps the pastry hold its shape in the oven.
Use the thermometer in the side of the Wellington so the temperature reading reflects the center.