Preheat the oven to 400 degrees F. and lightly grease 1 large baking sheet.
In a large heavy skillet or saute pan, heat the oil over medium-high heat. Add the onions and lower the heat. Cook slowly, stirring occasionally, until cooked down to a thick puree and very lightly caramelized, 35 to 40 minutes.
Add the garlic, salt, pepper, and thyme, and cook, stirring, for 1 minute. On a lightly floured surface roll out the pastry to a 16 by 12-inch rectangle. Transfer to the greased baking sheet. Wet the outer 1/4-inch edge with water and fold in to form a border. Very lightly prick the bottom of the pastry with a fork. Spread the onion filling evenly over the middle, being careful to leave the border uncovered. Arrange the anchovy fillets in a criss-cross pattern over the onions and decoratively arrange the olives around the squares. Bake until the bottom and edges are golden brown, about 20 minutes. Remove from the oven and serve warm, drizzled with extra-virgin olive oil.
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