Step 1

Beat 3/4 cup butter and and white sugar together in a bowl with an electric mixer until creamy. Beat egg into butter mixture.

Step 2

Whisk flour, cornmeal, baking powder, and salt together in a bowl. Gradually add flour mixture to butter mixture; stir. Beat vanilla into butter mixture. Form dough into a ball, wrap tightly in plastic wrap, and refrigerate until firm, at least 1 hour.

Step 3

Preheat oven to 350 degrees F (175 degrees C). Lightly grease baking sheets.

Step 4

Roll dough out onto a lightly floured surface to 1/8-inch thick. Cut dough with a round or Linzer cookie cutter. Use a smaller cutter to cut the center from the tops. Place cookie bottoms and tops 1-inch apart on prepared baking sheets.

Step 5

Bake in the preheated oven until edges are lightly golden, 10 to 12 minutes. Transfer to wire racks to cool.

Step 6

Stir cranberries, brown sugar, and water together in a saucepan over medium-high heat; cook until cranberries are soft, about 10 minutes. Stir butter and lemon juice into cranberry mixture; remove from heat and cool.

Step 7

Spread a small amount of the cranberry mixture onto one side of the bottom half of a cookie. Place the top half of the cookie on top of the cranberry mixture. Repeat assembly process with remaining cookies and cranberry filling.