Thaw the puff pastry according to package directions until cold but pliable, usually 30–40 minutes at room temperature.

Heat the oven to 400°F and line two baking sheets with parchment paper.

In a small bowl, stir together the goat cheese, thyme, and black pepper until combined.

Beat the egg with the water in a small bowl.

Working with one pastry sheet at a time, spread 1/4 cup fig jam over the pastry, leaving a 1/2-inch border along one long edge, then spread half the goat cheese mixture over the jam. Brush the bare edge with egg wash, roll the pastry tightly toward that edge, and place the log seam-side down; repeat with the second sheet.

Refrigerate the logs for 20–30 minutes, until firm enough to slice cleanly.

Cut each log into 1/2-inch slices with a sharp knife and arrange them about 1 inch apart on the prepared baking sheets.

Brush the tops with egg wash and bake for 15–18 minutes, swapping the sheets between oven racks halfway through, until the pastry is puffed and golden brown.

Cool the pinwheels on the baking sheets for 5 minutes before serving.