IN A LARGE HEAVY SKILLET, heat 2 tablespoons of oil over medium-high heat until hot but not smoking. Add the onions and saute them until just limp, about 3 to 4 minutes. Add the remaining oil, then stir in the mushrooms.
Adjust the heat to high and cook the mushrooms until the moisture has almost completely evaporated, 7 minutes, stirring often. Remove the pan from the heat, stir in the cheese, chervil and pepper, and set the filling aside while rolling out the pastry. Lightly flour a clean workspace.
Unfold the pastry and dust lightly with flour. Using a floured rolling pin, roll the pastry into a rectangle about 12-by-18 inches. Cut the rectangle into 2 long rectangles measuring 6-by-18 inches each.
Brush off the excess flour, and spread half of the mushroom mixture over each rectangle, leaving a 1-inch border at the uppermost edges. Starting with the bottom long edge, roll up each rectangle jelly roll fashion gently pressing the top edge to seal the pastry. Carefully wrap the rolls tightly in plastic wrap or foil.
Flatten each roll slightly by pushing gently on the sides so the spirals will be oval-shaped, and refrigerate until the pastry is firm, at least 2 hours. This may be done up to 4 days ahead of time. Preheat oven to 375F, and line a cookie sheet (or 2 if you have them) with baking parchment.
Remove the rolls from the refrigerator as you are ready to cut them. With a sharp knife, cut the rolls crosswise into 3/8-inch slices. Place each oval onto the parchment, leaving 1 1/2 inches of space between them.
Transfer the pan(s) to the middle of the oven, and bake until the tops are lightly browned. With a spatula, turn the spirals over and continue cooking until the second side is lightly browned and the centers are cooked through.
If using 2 pans, rotate the pans during baking for even baking. Remove the pans, slide the swirls onto a rack to cool, and continue in the same fashion until all the swirls are baked. Serve warm.
NOTE: If using your food processor to chop mushrooms, you will be far more successful if you pulse them several times to reach the desired consistency rather than let the machine run. Over-processed mushrooms are often watery and will not brown.