Preheat the oven to 375 degrees F (190 degrees C).
Beat butter, white sugar, and brown sugar together in a bowl with an electric mixer until smooth, 2 to 3 minutes. Add egg and vanilla extract; beat until combined. Add red wine; beat until fully incorporated. Stir flour, cocoa powder, baking soda, and salt into butter mixture until moistened. Scoop into small mounds on a baking sheet.
Bake in the preheated oven until edges are crisp, 10 to 12 minutes. Let cool on baking sheet for 10 minutes before transferring to a wire rack.