Pat the beef tenderloin dry and season generously with salt and pepper. Heat olive oil in a large skillet over high heat. Sear the tenderloin on all sides until a deep brown crust forms, about 30-60 seconds per side. Remove from the pan, let cool completely, then brush all over with Dijon mustard.

Pulse the mushrooms in a food processor until finely chopped. Sauté the minced shallot in the same skillet used for the beef over medium heat. Add the mushrooms and thyme, and cook, stirring occasionally, until all the liquid has evaporated. Season with salt and pepper and set aside to cool completely.

On a large piece of plastic wrap, lay out the prosciutto slices in a rectangle, overlapping them slightly. Spread the cooled mushroom duxelles mixture evenly over the prosciutto. Place the cooled beef tenderloin at one end. Tightly roll the beef in the prosciutto and duxelles using the plastic wrap to form a tight log. Chill for at least 30 minutes.

On a lightly floured surface, roll out the puff pastry sheet. Unwrap the chilled beef log and place it on the pastry. Brush the edges of the pastry with the beaten egg. Wrap the pastry around the beef log, trimming any excess and pressing the seam to seal it tightly.

Wrap the pastry-covered log in plastic wrap and chill in the refrigerator for at least 20 minutes to firm up. Once chilled, unwrap and use a very sharp knife to slice the log into 1-inch thick rounds.

Preheat your oven to 400°F (200°C). Place the bites on a parchment-lined baking sheet. Brush the tops and sides of each bite with the remaining egg wash and sprinkle with flaky sea salt. Bake for 15-20 minutes, or until the pastry is puffed and deep golden brown.