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For the filling: Preheat the oven to 375 degrees F. Place the tin on a sheet pan lined with parchment paper.
Lightly flour a flat surface. Unfold each sheet of puff pastry onto the floured surface and lightly roll out with a rolling pin to an even thickness. Brush both sheets with the butter leaving a 1-inch border on the puff pastry. Divide the blackberries over the butter.
Combine the granulated sugar and cinnamon in a small bowl and sprinkle over the surface of each sheet.
Starting with the end nearest you, roll the pastry up snugly like a jelly roll around the filling, finishing the roll with the seam-side down. Trim the ends of the roll about 1/2 inch and discard. Slice the roll in 6 equal pieces, each about 1 1/2-inches wide. Place each piece, spiral-side up, in 6 of the muffin cups. Repeat with the remaining pastry sheet.
Bake until the sticky buns are golden to dark brown on top and firm to the touch, 30 minutes. Remove from the oven onto a baking rack and let cool for 5 minutes. Use a small offset spatula to remove the buns from the tins, and place upside down on a serving platter.