Heat the oil in a large skillet over medium heat. Add the onion and mushrooms and saute until mushrooms have released their moisture and the onions have softened, about 10 minutes. Stir in the garlic and cook 2 minutes more. Season with salt and pepper. Stir in parsley and lemon zest.
Set a large sheet of parchment paper on a clean work surface and place the puff pastry on top of it. Roll pastry into a large ¼-inch-thick rectangle. Prepare it for braiding: Using a sharp knife, trim both corners off of one of the short sides; then cut two notches into the opposite side. Slice an equal number of 1-inch wide diagonal strips down both of the long sides of the pastry, leaving the center uncut. (For a video illustration of this process, watch here.)
Spoon the ricotta down the center of the pastry in a thin line. Top with the mushroom, onion, and herb mixture and sprinkle with Parmesan. Fold the short ends of the pastry over the filling like an envelope, then "braid" the top by folding the strips diagonally over the top, alternating sides as you go. (Again, for a visual guide, refer to the video above.) Using the parchment paper to support it, transfer the braided pastry to a baking sheet. Brush the top with beaten egg and bake until golden brown and crispy, about 25-30 minutes. Let cool slightly before slicing and serving.
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