Preheat the oven to 400 degrees F.

On a lightly floured surface, roll out the pastry to a rectangle about 8 by 12-inches. Fold the edges of the long sides over about 1 1/2-inches to make an edge on the tart. Score the edges lightly with a small knife, being careful not to cut all the way through the dough. Transfer to a baking sheet lined with parchment paper.

In a small bowl, mash the goat cheese with the honey. Spread the cheese evenly across the pastry inside the scored border. Arrange the pears and blackberries decoratively across the top of the cheese. Sprinkle evenly with the sugar and bake until the pastry is golden and the fruit is soft, 25 to 30 minutes. Remove the tart from the oven and transfer to a wire rack to cool before serving. In a bowl, whip together the cream and sour cream. Add the sugar and whip until soft peaks just start to form. To serve, cut the tart and serve the cream on the side.