Preheat oven to 350°F (175°C). Grease three large baking sheets. Cut three 10-inch rounds of parchment paper and set a round on each baking sheet.

Beat 1 cup butter, brown sugar, and white sugar together in a bowl using an electric mixer until fluffy and smooth. Mix in eggs, 2 teaspoons vanilla extract, baking soda, and salt; stir in flour until just combined. Fold chocolate chips into dough.

Divide dough into 3 parts; press each part onto a parchment round.

Bake until cookie rounds are browned and crisp, about 15 to 20 minutes. Remove from oven and cool completely.

Stir melted butter, milk, and 1 teaspoon vanilla in a bowl. Beat confectioners' sugar and cocoa in a separate bowl until combined. Mix butter mixture into cocoa mixture until frosting is smooth.

Peel parchment from cookies. Set aside the prettiest round for the top layer. Spread half of the frosting on one cookie, top with another cookie, spread remaining frosting, and add the last cookie layer.