Bring the strudel dough to room temperature for 1 hour before stretching, keeping it in the ziptop plastic bag until ready to use.
Preheat oven to 400 degrees and place a rack in the lower third of the oven. Line a baking sheet with parchment paper.
Heat the olive oil in a large heavy skillet over high heat. When the oil is shimmering, add the mushroom slices and reduce the heat to medium-high. Let them pan-roast and get deeply browned, 6 to 8 minutes (do not stir), before giving the pan a hard shake so they release. Then turn the mushrooms and roast for another 4 to 5 minutes, until well browned and crisped. Add the onion, stir well, and cook until wilted, 5 or 6 minutes. Add the chopped kale and cook for another 3 minutes, until the kale has collapsed but is still brightly colored. Stir in the parsley, thyme, salt and pepper. Spread the mixture across a rimmed baking sheet to cool. (Freeze or refrigerate to speed the cooling process.) Once thoroughly cool, return mushroom mixture to a bowl, and stir in the egg and cheese.
Spread a large, thin kitchen towel across a work surface. Stretch the strudel dough on the kitchen towel to 20 inches by 24 inches, until it's possible to "read a newspaper through it" or some close approximation of that idea. The whole process doesn't take long at all, just 5 minutes or so, once you've done it a few times.