Cut a piece of parchment to fit a rimmed baking sheet. Place 1 sheet of puff pastry on the parchment, then top with the second sheet of puff pastry. Roll the 2 layers to one 17-by-13-inch rectangle.
Combine the hot sauce and butter in a small bowl and brush both sides of the chicken cutlets with the sauce. Set aside.
Brush the puff pastry with the remaining sauce, leaving a 1-inch border along one long side and the 2 short sides. Sprinkle the blue cheese over the sauce. Top with 4 of the mozzarella slices, shingling them if necessary to fit. Top with 3 of the cutlets. Repeat with the remaining 4 slices of mozzarella and 3 cutlets.
Brush the 3 empty borders of puff pastry with the beaten egg. Fold over the long side without the egg wash, followed by the 2 short sides, then the other long side. Turn the Wellington over so it is seam-side down. Brush with the remaining egg and cut three 3-inch-long vents with a sharp knife on top.
Transfer the Wellington and parchment to the baking sheet and bake until crisp and deep golden brown, 45 to 50 minutes. Serve with ranch dressing, carrot and celery sticks, and more hot sauce.
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