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.