Toast the walnut halves in a dry 18 cm / 7-inch skillet over medium heat for 3 to 4 minutes, stirring often, until fragrant and lightly browned. Let cool completely.
Trim the stem cap off each dried persimmon with a paring knife.
Lay a persimmon on a cutting board and slice it open lengthwise on one side, cutting about three-quarters of the way through so it opens flat like a book without falling apart. Remove any seeds.
Press the persimmon flat with your palm to widen it, then place 2 toasted walnut halves at one end.
Roll the persimmon tightly around the walnuts from that end, using the fruit's own stickiness to seal the seam.
Wrap each roll tightly in plastic wrap, twisting the ends closed like a candy wrapper. Repeat with the remaining persimmons.
Chill the wrapped rolls for 30 minutes so they firm up and hold their shape when sliced.
Unwrap and slice each roll crosswise into 3 to 4 rounds with a sharp knife, wiping the blade clean between cuts.
Arrange the pinwheels cut-side up on a plate, dust lightly with cinnamon if using, and serve.