Place the olive oil and onions in a saucepan over a medium heat and cook for 20 minutes, stirring occasionally. Add the sugar, balsamic vinegar, salt, and pepper.
Cook the mixture for another 5 minutes, or until the onions are soft and caramelized. Remove from the heat and set aside to cool.
Preheat the oven to 425°F.
Roll out the pastry to an 8-by-16-inch rectangle on a lightly floured surface. Trim the edges with a sharp knife and place on a baking tray lined with parchment paper. Score a 1/4 inch border around the edge of the pastry, taking care not to cut all the way through to the bottom. Prick the pastry with a fork and sprinkle the Parmesan cheese within the scored edges.
Place the onion on top of the pastry base, again within the scored edges. Bake in the oven for 20 to 25 minutes, or until the pastry is puffed and golden brown. Remove from the oven and sprinkle with the feta cheese and oregano leaves.
Serve warm with a crispy salad.
Beverage pairing: Château de Montfort Vouvray, France. Vouvray, made from the Chenin Blanc grape, has the perfect blend of sweet fruit and earthy flavors to complement the sugars that emerge from cooking onion. In addition, it has a touch of piquant acidity, so the feta cheese won’t bowl it over.
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