Preheat the oven to 220 °C with a large baking sheet (or pizza stone) in the oven. The pan will be placed on top of this baking sheet later.
Start caramelizing the onions in the pan: Cut the onion vertically into slices about 6 mm (1/4-inch) thick.
In a wide oven-safe pan (mine was 27 cm (10.6 inch) diameter), melt the butter over medium-low heat, add the onions and the sugar. Cook for 15 minutes, stirring from time to time to help avoid burning the onions. You may also add a spoonful of water as necessary to avoid burning if the pan dries out.
Add the other ingredients to the veg: Add the salt, black pepper, dried thyme, caraway seeds, and dried figs. Cook another 5 minutes or until the onions have taken on a nice golden color, but not totally caramelized.
Stir in the chopped olives. Crumble in the fresh goat cheese. Don’t stir too much at this point to keep the goat cheese chunks intact.
Put on the pastry: Lay the pastry over the vegetables in the pan, and tuck in the edges. Cut 4 little slits in the top of the dough to let steam escape.
Bake: Place the pan in the oven on top of the baking sheet, and bake in the lower half of the oven at 220 °C for 25 minutes, or until the crust is golden brown.
Take the pan out of the oven, and let rest for 5 minutes before turning out onto a large plate or cutting board. (See note 3)