Heat the butter in a medium skillet over medium heat. Add the mushrooms and thyme and cook, stirring occasionally, until all the moisture has evaporated from the mushrooms, 12 to 15 minutes (make sure to cook out all the moisture or your pastry will not become crisp). Add the sherry vinegar and cook 1 more minute. Season with the kosher salt and pepper and let cool completely.
Lay out a few sheets of paper towels and pat dry the filet cubes. Heat the olive oil in a large cast-iron pan over high heat. In batches if necessary, sear all sides of the meat cubes, about 45 seconds per side. Remove from the pan and let rest until cool, then brush on all sides with the Dijon mustard.
On a baking sheet, lay out a half slice of prosciutto and spread with a thin layer of the mushroom mixture. Place a filet cube in the center and wrap the prosciutto around it. Repeat with the remaining prosciutto, mushroom mixture and filet cubes. Wrap the baking sheet with plastic wrap and pop it in the freezer for 15 minutes.
Preheat the oven to 400 degrees F. Line a second baking sheet with parchment paper.
Cut sixteen 2-inch squares from the pastry. Put a prosciutto-wrapped filet cube in the middle of each puff pastry square. Bring the corners together and seal, using water if necessary to get the pastry to stick together.
Place the Wellingtons seam-side down on the lined baking sheet. Beat the egg with 1 teaspoon water, then brush the pastry with the egg wash. Sprinkle with some sea salt. Bake until golden, about 5 minutes.