Preheat oven to 350 degrees F (175 degrees C). Grease 3 large baking sheets. Cut three 10-inch rounds of parchment paper and set each 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 eggs, 2 teaspoons vanilla extract, baking soda, and salt into creamed butter until just combined; stir in flour until dough is just combined. Fold chocolate chips into dough.
Divide dough into 3 parts; press 1 part onto each parchment round.
Bake in the preheated oven until cookie rounds are browned and crisp around the edges and set in the middle, 15 to 20 minutes. Remove from oven and cool completely.
Stir melted butter, milk, and 1 teaspoon vanilla extract together in a bowl. Beat confectioners' sugar and cocoa powder together in a separate bowl until evenly combined. Beat butter mixture into cocoa mixture until frosting is smooth and thickened.
Peel parchment paper off the bottom of each cookie round. Set aside the prettiest round to be used for the top layer. Spread 1/2 the frosting onto 1 cookie round and top with a second cookie round. Spread remaining frosting onto second cookie round and top with the third cookie round.
