Make the pate brisee and roll into a cylinder. Chill for 1/2 hour or up to 3 days.
Heat the butter and 1 tablespoon of the oil in a large skillet. Season with salt, pepper and thyme and cook the onions over medium low heat until caramelized. Remove the onions from skillet and set aside in a bowl to cool. Heat the remaining tablespoon of oil in the skillet and saute the potatoes, seasoned with salt and pepper, until about 3/4 cooked. Combine the potatoes with the onions, mix in 1/4 cup Parmesan and set aside to cool completely. Preheat oven to 400 degrees.
Cut the cylinder of dough in half and refrigerate one half. Press the piece of dough into a rectangle and then roll it, on a lightly floured surface, into a rectangle measuring about 17 x 8 inches. Brush off any excess flour and transfer the dough to an ungreased cookie sheet. Position it so that there is room for the second piece. Refrigerate the tray while you roll out the second piece of dough in the same way and place it on the cookie sheet.
Using a serrated pastry wheel, trim the edges of the rectangles to even them. Cut a "v" out of each corner to prevent the dough from bunching when it's folded. Spread half the potato/onion mixture on each, leaving a 1 1/2-inch border all around. Fold up the edges. Brush the dough with egg wash and sprinkle the remaining Parmesan over the dough and filling. Bake about 40 minutes, or until the crust is nicely browned. Slice each section on the diagonal into about 6 pieces.
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