Beat together the butter and sugar in a large bowl with an electric mixer until fluffy. Beat in 1 teaspoon of the vanilla and the egg until well blended, then the zest.
In a separate bowl, stir together the flour, baking powder, baking soda, and salt. Gradually mix the dry ingredients into the butter mixture until just combined. Shape the dough into a disk, wrap in plastic, and chill for at least 1 hour, until firm.
Make the filling while the dough chills. Combine the figs and orange juice in a small saucepan and bring to a simmer over medium heat. Cook until the figs are soft, 5 to 15 minutes, depending on how dry the figs are. Drain and purée with the remaining 1/2 teaspoon vanilla. Set aside to cool.
Heat the oven to 375°F. On a lightly floured work surface, divide the dough into quarters and roll each piece into a long rectangle about 4 inches wide; square off the edges of the dough as needed to keep it relatively even. Spoon the fig filling lengthwise evenly down the middle of each piece of dough; fold up the sides of the dough around it and press gently to seal. Bake on ungreased cookie sheets, seam side down, for about 15 minutes, until the logs are evenly golden. Cut into squares while they are still warm. Store in a tightly covered container for up to 1 week.