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Heat oil in a 12-inch heavy skillet over medium heat until it shimmers, then cook onions with 1/2 tsp salt and 1/4 tsp pepper, covered, stirring occasionally, until golden, 15 to 20 minutes. Transfer to a small bowl and keep warm, covered.
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Put a large heavy baking sheet (17 by 14 inches) on lowest rack of oven, then preheat oven to 500°F.
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Meanwhile, roll out dough on a lightly floured surface with a floured rolling pin, stretching corners with your hands to form a 16- by 13-inch rectangle. (Dough will be easier to roll out as it warms.) Transfer to a large tray lined with sheet of parchment paper. Lightly prick dough all over with a fork, then slide dough (on parchment) from tray onto hot baking sheet.
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Bake until top is puffed and pale golden in patches, about 10 minutes. (Prick any large bubbles with a fork and flatten.) Remove crust from oven and leave oven on.
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Sprinkle crust evenly with cheese, walnuts, and parsley. Bake pizza until edge of crust is deep golden, cheese is bubbling and golden in patches, and underside is golden brown, 2 to 4 minutes. Serve immediately.
