Special equipment: A baking stone or baking sheet
Position an oven rack in the bottom position of the oven, put a baking stone or baking sheet on it and preheat to 400 degrees F. Line another baking sheet with parchment.
Dust a work surface with flour, and roll the puff pastry sheet out into a 10-by-15-inch rectangle. Transfer the pastry to the prepared baking sheet. Measure a half-inch border around it, and lightly score the border with a knife (don't cut all the way through). Prick the pastry inside the border all over with a fork (don't prick the border). Brush the pastry all over with the egg, put the baking sheet with the pastry on it on top of the preheated baking stone, and bake until golden brown, 15 to 20 minutes. Let cool on the baking sheet for a few minutes. If you find that the pastry has puffed up in some places, use the bottom of a measuring cup or glass to press it down. Let the pastry cool completely on the baking sheet. (The cooled pastry can be wrapped and stored at room temperature for 1 day.)
Meanwhile, heat the oil in a medium skillet over medium-low heat. Add the onions and a large pinch of salt, and cook, stirring occasionally, until soft and deeply golden, about 10 minutes. Add the vinegar, and stir until it has been completely absorbed. Remove the skillet from the heat, and let the caramelized onions cool completely. (They can be refrigerated for 1 day.)
Scatter the fontina all over the pastry within the border. Top with the caramelized onions, figs and walnuts. Sprinkle the whole tart with the feta, thyme and some salt and pepper. Drizzle all over with oil.
Return the tart on the baking sheet to the oven, putting it on the preheated baking stone. Bake until the fontina melts, 10 to 15 minutes. Drizzle generously with honey, and let cool for a few minutes. Cut into 16 squares, and serve.
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