Make the filling: In a food processor, combine dried sour cherries, jam, and sugar. Process until coarsely pureed. Transfer to a small saucepan over medium heat. Cook, stirring occasionally until it comes to a boil. Remove from heat, and stir in almond extract. Let stand until cool.
Make the dough: In a medium bowl, whisk together flour, cornmeal, baking powder, and salt; set aside.
In the bowl of an electric mixer fitted with the paddle attachment, cream butter and sugar on medium speed until light and fluffy, 2 to 4 minutes. Add egg and vanilla; incorporate. With mixer on low speed, add flour mixture to butter mixture; mix to combine.
Transfer dough to a clean work surface, and divide into four equal portions. Place one portion between two pieces of parchment paper that are at least 12 inches square. Roll out dough to a 3 1/2-by-9-inch rectangle about 1/4 inch thick. Repeat with the remaining three portions dough. Transfer to a baking sheet and freeze for about 30 minutes.
Assemble the cookies: Remove dough from freezer. Remove top pieces of parchment from dough. Spread one-third of the filling evenly over one portion of dough. Invert second rectangle over the first; remove parchment on top. Repeat layering process, leaving the top rectangle uncoated. Trim to a 3 1/4-by-8 1/2-inch brick. Wrap in plastic wrap and transfer to freezer for 1 hour.
Preheat oven to 350 degrees. Line baking sheets with Silpats (French nonstick baking mats). Cut the brick into 1/4-inch-thick rectangles; place on baking sheets, spaced 2 1/2 inches apart.
Bake until light golden around the edges, 12 to 15 minutes. Transfer to a wire rack to cool. Store in an airtight container for up to 2 weeks.