Preheat oven to 375°F with two racks near the center.
Unfold each sheet of puff pastry onto a lightly floured cutting board. Roll it out just slightly with a floured rolling pin.
Using a 5-inch cookie cutter (or an overturned bowl and a small, sharp knife), cut four 5-inch circles from each sheet of dough.
Divide the rounds between two parchment-lined baking sheets.
Press a 3-inch cookie cutter into the center of each round, stopping ¾ of the way down so that the outer and inner portions are still attached at the bottom. This will allow the outer ring to rise around the center. Pierce the inner section of each round all over with a fork.
Place the goat cheese, powdered sugar, lemon juice, and salt in a small bowl. Microwave for 30 seconds or until soft but not melted. Stir to combine well.
Divide goat cheese mixture evenly among pastry rounds and spread over the inner section of each round.
Top with overlapping apple slices, making sure to stay within the inner section of each round.
Combine the butter and honey in a small bowl and microwave for about 30 seconds, watching carefully, until the butter is almost melted. Stir to combine and complete the melting. (Be careful, as the honey gets hot surprisingly quickly.)
Brush some honey butter over the entirety of each round.
Bake for about 25 minutes, until the apples are tender and the pastry is puffed and nicely browned.
Let the pastries cool slightly, then dust lightly with ground allspice and drizzle with a bit of additional honey.
Serve warm.