In a food processor, pulse the mushrooms, shallot, garlic, and thyme until very finely chopped. Heat a large, dry skillet over medium heat. Add the mushroom mixture and cook, stirring occasionally, until all the moisture has evaporated, about 15-20 minutes. Season with salt and pepper, remove from heat, and let cool completely.

Pat the beef tenderloin cubes dry and season generously with salt and pepper. Heat olive oil in a skillet over high heat until just smoking. Sear the cubes on all sides until well-browned, about 30-45 seconds per side. The inside should still be rare. Remove from skillet and let cool completely. Once cool, brush all sides with Dijon mustard.

Preheat oven to 400°F (200°C) and line a baking sheet with parchment paper. On a lightly floured surface, unfold the puff pastry sheets. Cut each sheet into 12 squares (about 3x3 inches). Place a small piece of prosciutto on each square. Top with about a teaspoon of the cooled duxelles. Place a seared beef cube in the center. Gently pull the four corners of the pastry up and pinch them together to enclose the beef. Place seam-side down on the prepared baking sheet.

Brush the assembled bites with the egg wash and sprinkle with flaky sea salt. Place the baking sheet in the refrigerator to chill for at least 15 minutes. Bake for 15-20 minutes, or until the pastry is puffed and deep golden brown. Let rest for 5 minutes before serving.