For the filling, combine the dried figs, water, honey, lemon juice, and cinnamon in a small saucepan. Bring to a simmer over medium heat and cook for 8 to 10 minutes, stirring occasionally, until the figs are very soft and most of the liquid is absorbed. Transfer to a food processor and pulse until a thick, smooth paste forms. Set aside to cool.
For the dough, whisk together the gluten free 1-to-1 baking flour, almond flour, brown sugar, baking powder, and salt in a large bowl. Add the cold butter and work it in with your fingers or a pastry cutter until the mixture resembles coarse crumbs. Stir in the egg and vanilla extract until a soft dough forms. Wrap in plastic wrap and chill for 30 minutes.
Preheat the oven to 350 F and line a baking sheet with parchment paper. Divide the dough in half. Roll one half between two sheets of parchment paper into a rectangle about 12 inches long and 4 inches wide. Spread half the fig filling down the center in a long strip. Fold the dough edges up and over the filling, overlapping slightly, and press gently to seal. Flip seam-side down onto the baking sheet. Repeat with the remaining dough and filling.
Bake for 18 to 22 minutes until lightly golden. Let cool on the baking sheet for 15 minutes.
Slice each log crosswise into 1-inch bars using a sharp tool. Let cool completely before serving.