To prepare the cookies: In a medium bowl, beat butter with an electric mixer on medium to high speed for 30 seconds. Add granulated sugar, baking powder, and salt. Beat until combined, scraping side of bowl occasionally. Beat in oil, egg, and vanilla until combined. Beat in as much of the flour as you can with the mixer. Using a wooden spoon, stir in any remaining flour and the cornmeal. Divide dough in half. Wrap and chill dough for 2 to 3 hours or until easy to handle.

Preheat oven to 375 degrees F. On a lightly floured surface, roll half of the dough to an 1/8-inch thickness (keep remaining dough chilled until ready to use). Using a 1-1/2- to 2-inch cookie cutter, cut into desired shapes. Using a 3/4- to 1-inch cutter, cut out the center of half of the cutouts. (Reroll the small cutouts and the scraps as necessary.)

Arrange cutouts 1 inch apart on ungreased cookie sheets. Bake about 5 minutes or just until edges are firm (cookies will not brown). Transfer cookies to a wire rack; cool. Repeat with remaining dough.

To prepare the Raspberry-Fig Filling: in a small saucepan, combine orange juice, raspberries, and figs. Bring to boiling; reduce heat. Simmer, uncovered, for 5 to 7 minutes or until mixture is thickened, stirring frequently. Remove from heat. Transfer mixture to a blender or food processor. Cover and blend or process until nearly smooth.

Sprinkle the tops of the cookies that have the centers cut out with powdered sugar; set aside. Spread Raspberry-Fig Filling onto each whole cookie. Top with cookies with cutout centers.