Combine chopped figs, water, granulated sugar, orange juice, orange zest, cinnamon, and nutmeg in a medium saucepan over medium heat. Cook, stirring occasionally, for 8–10 minutes until the figs are very soft and the mixture is thick and jammy.
Transfer the cooked fig mixture to a food processor and pulse until a smooth, thick paste forms. Scrape into a bowl, cover with plastic wrap pressed directly onto the surface, and cool completely before using.
Whisk together the flour, baking powder, and salt in a medium bowl; set aside.
Beat the softened butter and granulated sugar together on medium speed for 3–4 minutes until pale and fluffy. Add the egg and vanilla extract and beat until fully incorporated.
Add the flour mixture to the butter mixture in two additions on low speed, mixing just until no streaks of flour remain. Do not overmix.
Shape the dough into a flat disc, wrap tightly in plastic wrap, and refrigerate for at least 30 minutes. Preheat oven to 350°F (175°C) and line baking sheets with silicone mats or parchment paper.
On a lightly floured surface, roll the chilled dough into a rectangle approximately 12 x 10 inches and ⅛ inch thick.
Spread the cooled fig filling evenly over the dough, leaving a ½-inch border along all edges.
Starting from one long edge, roll the dough into a tight log, pinch the seam closed, and wrap tightly in plastic wrap. Refrigerate for 15–20 minutes until firm.
Using a sharp knife, cut the chilled log into ½-inch slices with one clean downward motion per slice. Place cookies cut-side up on prepared baking sheets, spaced 2 inches apart.
Optionally brush the tops with milk, then bake for 12–14 minutes until the edges are lightly golden and the tops look set. The centers will appear slightly underdone — that's correct.
Let cookies rest on the pan for 5 minutes, then transfer to a wire cooling rack to cool completely. Dust with powdered sugar if desired before serving.