In a large 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 nicely browned, sprinkle with salt and pepper to taste and set aside to cool to room temperature.
Preheat the oven to 375℉. Line two 14x17-inch sheet pans with parchment paper and transfer one sheet of pastry dough to each. Roll out the 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 the parchment). Slightly separate the 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 ½ teaspoon of the blue cheese.
Bake in oven until fully puffed and golden, about 25 to 30 minutes. Serve warm.
Notes
Depending on how you cut the puff pastry, you could get anywhere from 36 to 48 appetizers.