Pat the beef cubes dry with paper towels and season generously with salt and pepper. Heat 1 tbsp olive oil in a large skillet over high heat. Sear the beef cubes for 30-60 seconds on each side until browned. Do not overcrowd the pan. Transfer to a plate and let cool completely in the refrigerator.

In the same skillet, add the remaining 1 tbsp olive oil. Sauté the shallot until softened, about 2 minutes. Add the finely chopped mushrooms and thyme. Cook, stirring occasionally, for 8-10 minutes, until all the moisture has evaporated and the mixture is dry. Season with salt and pepper. Set aside to cool completely.

On a clean work surface, lay out a slice of prosciutto. Spread a thin layer of Dijon mustard on it, followed by a thin layer of the cooled mushroom duxelles. Place one cooled beef cube in the center and wrap the prosciutto around it.

Preheat your oven to 400°F (200°C) and line a baking sheet with parchment paper. Unfold the puff pastry sheets and cut each into 12 squares (about 3x3 inches). Place a prosciutto-wrapped beef cube in the center of each pastry square. Bring the four corners of the pastry up to the center and pinch the seams together firmly to enclose the beef completely.

Place the bites seam-side down on the prepared baking sheet. Brush the tops and sides with the egg wash and sprinkle with flaky sea salt. Bake for 15-20 minutes, or until the pastry is golden brown and puffed. Let them cool for a few minutes before serving.