Preheat the oven to 350 degrees F (175 degrees C). Grease 3 large baking sheets; place one 10-inch round parchment paper on each baking sheet.

Beat 1 cup butter, brown sugar, and white sugar together in a bowl with an electric mixer until fluffy and smooth; beat in eggs, 2 teaspoons vanilla extract, baking soda, and salt until just combined. Stir in flour until dough forms; fold in chocolate chips.

Divide dough into 3 equal parts; press 1 part onto each parchment paper round and shape into large cookies.

Bake in the preheated oven until cookies are brown and crisp around edges and set in the middle, 15 to 20 minutes. Cool completely, about 1 hour; discard parchment paper.

Combine 1/2 cup melted butter, milk, and 1 teaspoon vanilla extract in a bowl. Whisk confectioners’ sugar and cocoa powder together in a separate bowl until combined; beat in butter mixture until frosting is smooth and thick.

Set aside prettiest cookie round for top layer. Spread 1/2 frosting on 1 cookie round; top with 1 cookie round. Spread remaining 1/2 frosting on second cookie round; top with remaining (prettiest) 1 cookie round.