If you'll be preparing these immediately, heat the oven to 400 degrees and line a large baking sheet with parchment.
Roast the squash at 400 degrees in the oven with the olive oil, salt, and pepper - it will take about 20 minutes. You can stir in the syrup, thyme, and paprika when you take it out of the oven. I did this the day before and stored it in a container in the refrigerator.
While the squash is roasting, put a large skillet on medium and melt the remaining butter in the skillet. Stir in garlic and shallot; cook 1 to 2 minutes. Add the mushrooms and remaining salt. Cook until mushrooms are soft and their juices evaporate, about 10 minutes. Stir in the wine and cook until the mixture is dry, about 5 minutes. Stir in the pepper and parsley. Taste and add more salt if needed. You can also do this the day before, if you like.
Unwrap the puff pastry and place one on a piece of parchment or silicone silpat-style mat that you can transfer directly to your sheet pan for baking. Ms. Clark cuts her puff pastry sheets into into two 5-by-15-inch rectangles. I do not cut the puff pastry. I wanted a larger finished wellington that I could slice into pieces. Using a fork, poke holes in the pastry to let it steam and create crispy layers (less puff - more crisp layers).
Spread mushrooms on each pastry rectangle leaving 1/4-inch border. Spoon the cheese crumbles over the mushrooms. Then spoon the squash over the cheese, leaving a 1 1/2-inch border (it will look like a stripe of squash lying on a bed of cheese and mushrooms). It also works to put the squash on the mushrooms and the cheese last - just be sure to start with the mushrooms.
Brush the exposed borders of dough on each rectangle with the egg wash. Fold the long sides up to meet in the middle and pinch together to seal; pinch the ends, too. Repeat with other piece of puff pastry.
Transfer the parchment or silicone mat with pastry on top to the baking sheet. Turn them over so that the seam is face down. If you end up with any small holes in the puff pastry, just close them up with your hands. There isn't any liquid, so a tiny hole won't destroy the wellington. Brush the tops with more egg wash. Bake until they are puffed golden, and firm to the touch, about 30 minutes.* Let cool for 10 minutes, slice and serve.