Finely chop the cremini mushrooms (or pulse in a food processor) and mince the shallots and garlic.
Heat 2 tablespoons olive oil and the unsalted butter in a large skillet over medium heat. Add the mushrooms, shallots, fresh thyme leaves, and a pinch of kosher salt. Cook, stirring often, until the mixture releases its liquid and turns into a thick, dry paste, 12–15 minutes.
Stir in the Madeira wine and cook until completely evaporated and the duxelles is dry and sizzling, 4–6 minutes. Season to taste with kosher salt and black pepper, spread on a plate to cool quickly, and refrigerate until cold, 20–30 minutes.
Pat the beef tenderloin (center-cut) dry and season all over with 1 teaspoon kosher salt and 0.5 teaspoon black pepper.
Heat the remaining 1 tablespoon olive oil in a heavy skillet over medium-high heat. Sear the beef on all sides, including the ends, until well browned, 6–8 minutes total. Transfer to a rack to cool 5 minutes.
Brush the warm beef evenly with the English mustard, then refrigerate until just cool and tacky, about 15 minutes.
Lay a large sheet of plastic wrap on the counter. Shingle the prosciutto into a tight rectangle (slightly larger than the beef). Spread the chilled mushroom duxelles evenly over the prosciutto.
Place the beef at the near edge. Use the plastic wrap to roll the prosciutto and duxelles tightly around the beef, creating a uniform log. Twist the ends of the wrap to secure and chill until firm, 30 minutes.
Beat the large egg with the water to make an egg wash.
Dust the work surface with the all-purpose flour. Roll the puff pastry (all-butter) to about 1/8 inch thick, forming a rectangle that will fully encase the beef (about 12Ă—14 inches).
Unwrap the beef log and place it along the lower third of the pastry. Brush the pastry edges with some egg wash. Roll the pastry over the beef to enclose, overlapping the seam by about 1 inch. Trim excess, press to seal, and tuck the ends under. Transfer seam-side down to a parchment-lined baking sheet.
Brush the entire pastry with egg wash. Use the back of a knife to lightly score a decorative pattern without cutting through. Refrigerate 15 minutes to firm.
Preheat the oven to 400°F. Bake until the pastry is deep golden brown and an instant-read thermometer inserted into the center of the beef reads 120–125°F for medium-rare (or 130–135°F for medium), 30–40 minutes. If the pastry browns too quickly, tent loosely with foil.
Rest the Beef Wellington 10–15 minutes before slicing with a sharp serrated knife. Serve immediately.