In a 12-inch frying pan over medium heat, melt the butter with the olive oil. Add the sliced onion and reduce heat slightly (between medium and medium-low). Cook, stirring occasionally, until the onions soften and turn brown, 30 to 40 minutes. Meanwhile, trim the pear quarters to remove seeds, then cut into thin slices; set aside. When onions are done, sprinkle with salt and pepper to taste and set aside to cool to room temperature.
Preheat oven to 375° (convection oven not recommended). Line two 14- by 17-inch sheet pans with parchment paper and transfer one sheet of pastry dough to each. Roll out pastry lightly with a floured rolling pin to flatten any creases.
Using a pizza cutter or sharp knife, cut the pastry sheets into triangle, wedge, or diamond-shaped pieces about 2 inches wide (don't worry if you cut through parchment). Slightly separate pieces so that they are not touching.
Lay a pear slice in the center of each wedge (trim the slice so that it doesn't cover the entire pastry). Top with a small pile of caramelized onions and about 1/2 teaspoon of the gorgonzola or blue cheese.
Bake in oven until fully puffed and golden, about 25 to 30 minutes. Serve warm.
Do-ahead tips: Cook the onions up to 2 days ahead. Store onions in an airtight container in the refrigerator. Slice pears, prepare puff pastry, and bake just before serving.
Party short-cut: Hosting a bigger crowd? Buy extra puff pastry and get creative with toppings. Try sliced mushrooms and fontina cheese, or thinly sliced cooked red potatoes, thyme, and sea salt.
Servings Makes 36 to 48 pieces
* Percent Daily Values are based on a 2,000 calorie diet. Your daily value may be higher or lower depending on your calorie needs.