Step 1

Working in two batches, pulse the mushrooms in a food processor until finely chopped, 10 to 15 pulses. Heat 3 tablespoons butter in a large skillet over medium-high heat. Add the shallots and thyme and cook until softened, 1 to 2 minutes. Stir in the mushrooms and cook, stirring often toward the end, until dry and starting to form small crumbles, 25 to 28 minutes.

Step 2

Remove the skillet from the heat and add the cognac. Carefully return the skillet to the heat (the cognac may ignite) and continue cooking until the mushroom mixture is dry and the alcohol cooks off, 2 to 3 minutes. Stir in the remaining 1 tablespoon butter and season with kosher salt and pepper. Remove from the heat and let cool completely.

Step 3

Meanwhile, season the beef all over with kosher salt and pepper. Heat the vegetable oil in a separate large skillet over medium-high heat. Add the beef and cook, turning, until browned on all sides (including the ends), about 8 minutes. Transfer the beef to a rack and let cool.

Step 4

Overlap 2 sheets of plastic wrap on a work surface to make a large rectangle (about 22 by 30 inches) with a short side in front of you. Arrange the prosciutto slices crosswise on the plastic wrap in overlapping rows to form a rectangle slightly longer than the beef and wide enough to completely wrap it with a slight overlap. (You might not use all of the prosciutto.) Spread the mushroom mixture over the prosciutto, pressing gently. Cut the ties off the beef and brush it all over with the mustard. Place the beef on top of the prosciutto-mushroom mixture, perpendicular to the prosciutto slices. Tightly roll the prosciutto-mushroom mixture around the beef, using the plastic wrap to help you roll. Twist and tie the ends of the plastic wrap to help the roast hold an even shape. Refrigerate at least 30 minutes.

Step 5

Roll the puff pastry into a 14-by-15-inch rectangle on a lightly floured surface; if using 2 smaller sheets of pastry, press the ends together before rolling out, then trim as needed. Brush the pastry all over with the beaten egg. Remove the plastic wrap from the beef and place the beef on the pastry. Carefully roll the pastry around the beef, overlapping the ends at the seam; trim off any excess pastry, if needed, then pinch the seams together. Turn seam-side down. Fold in the pastry on the two open ends, trimming off any excess. Refrigerate until the pastry is cold and firm, about 1 hour.

Step 6

Position a rack in the lower third of the oven; preheat to 425˚ F. Line a baking sheet with parchment paper or foil. Place the beef seam-side down on the baking sheet and brush all over with the beaten egg. Score lines in the pastry with the dull side of a paring knife (don’t cut through the pastry) and sprinkle with flaky salt. Bake until the pastry is crisp and dark golden brown and a thermometer inserted into the center of the beef registers 110˚ F for rare to 120˚ F for medium rare, 40 to 50 minutes. Transfer to a rack set on a baking sheet and let rest 15 minutes before slicing. Sprinkle with more flaky salt and chives.