Heat the olive oil in a large heavy saucepan over medium-high heat. Add the onion and cook until translucent, about 5 minutes. Add the garlic and cook for 30 seconds until aromatic. Add the chicken broth, bell pepper, honey, ginger, vinegar, and chipotle chile, and bring to a boil. Reduce the heat to medium-low and simmer until the sauce thickens, about 2 1/2 hours. Add the raspberries and cook, stirring occasionally, for 15 minutes. Remove the pan from the heat and let the chutney cool slightly. Stir in the cilantro and season with salt and black pepper, to taste. (Chutney can be prepared 3 days in advance. Cool and store in the refrigerator).
Preheat the oven to 400 degrees F. Line a baking sheet with parchment paper.
On a lightly floured surface, roll the puff pastry into a 12-inch square. Cut the pastry into 2 to 3-inch squares. Put the squares on the prepared baking sheet. Using a small cookie cutter score a circle in the center of each square or using a paring knife, score a 1/2-inch border around the edge of each square. Bake the squares until puffed and golden, about 12 minutes.
Remove the scored, top section of the cooked pastry and fill with some of the goat cheese. Spoon a generous dollop of the chipotle-raspberry chutney on top of the goat cheese. Arrange on a serving platter and serve.