Preheat the oven to 400°F.
Roll out the puff pastry on a lightly floured surface and transfer it to a baking sheet lined with parchment paper.
In a small bowl, mix together the goat cheese, heavy cream, honey, and thyme until smooth.
Spread the goat cheese mixture over the puff pastry, leaving a 1-inch border around the edges.
Arrange the blackberries on top of the goat cheese mixture.
Brush the edges of the puff pastry with the beaten egg.
Bake the tart for 30-35 minutes, or until the pastry is golden brown and the filling is set.
Let the tart cool for a few minutes before slicing and serving.