Preheat the oven to 425 degrees F.
Cut a piece of parchment to fit a baking sheet. Stack both sheets of puff pastry on top of each other on top of the parchment and roll to a 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.
Brush the puff pastry with the remaining sauce, leaving a 1-inch border along both short sides and one long side. Sprinkle the sauce area with the blue cheese. Make a line with 4 of the mozzarella slices, slanting them if necessary to keep them within the sauce area. Place 3 of the cutlets on top of the mozzarella. 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 first, 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 vents for steam 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, more hot sauce and carrot and celery sticks.
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