Beat 3/4 cup butter and and white sugar together in a bowl with an electric mixer until creamy. Beat egg into butter mixture.
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.
Preheat oven to 350 degrees F (175 degrees C). Lightly grease baking sheets.
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.
Bake in the preheated oven until edges are lightly golden, 10 to 12 minutes.
Transfer to wire racks to cool.
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.
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.
