Clean mushrooms with damp cloth, stem and chop about ¼ inch.
For the duxelles, heat a large skillet over medium-high heat and melt butter to foam. Add mushrooms and brown until fragrant and volume reduces by about half, then add shallots, garlic, thyme, salt and pepper, and stir until shallots are softened, 3 minutes. Add sherry and let it absorb fully, then remove from heat and cool mixture in shallow bowl or on a tray.
Bring meat to room temp and pat dry. Gently press down each steak to spread surface area a bit, leaving steaks 2 to 2¼ inches thick, and season with salt and pepper. Heat a cast-iron skillet or griddle over high, add oil and brush or wipe it around the pan or spray with non-aerosol cooking spray). Add meat to pan, reduce heat to medium-high and cook to develop a crust on the meat, but not to cook through; it should remain very rare, 1½ to 2 minutes on first side, 1 on opposite. Turn off heat, add brandy and let it absorb, turning meat in it. Remove meat to a plate. Let cool completely, 10 to 15 minutes, then add about 1 teaspoon mustard to each steak and spread to coat only the top side.
Preheat oven to 400˚F, with rack at center.
Roll out dough a bit on floured surface and cut into 4 portions. Lightly roll out each puff pastry rectangle 3 to 4 times with a rolling pin until each rectangle is an even thickness (about 1/8 inch thick). Leave a border on each piece of dough and spread each piece with a centered layer of about 2 ounces mousse pate (do not press, just gently spread). Top with about one-fourth of cooled duxelles and center a steak, mustard-side down, on the mushrooms. Wrap and trim the pastry, then use egg wash to seal; flip and brush with more egg wash. Scraps can be used to make decorations to apply to crust if you have the patience for it.
Arrange Wellingtons on parchment-lined baking sheet and bake until golden.
