In a food processor, pulse flour, sugar, salt and half of cinnamon until mixed. Pulse in butter until coarse crumbs form. Pulse in egg and 2 Tbsp. juice until dough forms a ball.
Divide dough in half; pat into 2 4 1/2-inch-wide disks. Wrap disks in plastic wrap; chill for 1 hour.
In processor, puree figs, dates, honey and remaining juice and cinnamon until smooth paste forms. Transfer to a bowl; cover.
Unwrap 1 dough disk. Place between sheets of plastic wrap and roll out into an 8-by-10-inch rectangle, about 1/8 inch thick. Trim edges. Remove top sheet; spread half of filling over dough, leaving a 1/4-inch border. Starting with a short side, using plastic as an aid, roll up into a log. Wrap in plastic wrap and freeze until firm, at least 2 hours and up to 1 week. Repeat with remaining dough and filling.
Place racks in top and bottom thirds of oven; preheat to 350ºF. Line 2 large baking sheets with parchment. With a serrated knife, cut 1 log into 1/3-inch-thick slices. Place on sheets, 1 inch apart. Bake, switching sheets halfway through, until golden brown, 20 to 25 minutes. Let cool for 2 minutes; transfer to wire racks to cool completely. Repeat with remaining log.