Preheat oven to 350 degrees F (175 degrees C). Grease a 10 1/2x15 1/2-inch baking pan and line with waxed paper.
Beat 1 cup butter, brown sugar, and white sugar together in a bowl using an electric mixer until smooth and creamy; add eggs, 1 at a time, beating well after each addition until smooth. Mix 2 teaspoons vanilla extract and salt into creamed butter mixture. Gradually stir flour into creamed butter until just incorporated; fold 1 cup chocolate chips into batter. Spread batter into the prepared pan.
Bake in the preheated oven until lightly browned on the edges, 20 to 25 minutes. Remove from oven and cool completely.
Melt remaining 1 cup chocolate chips in the top of a double boiler over simmering water, stirring frequently and scraping down the sides with a rubber spatula to avoid scorching. Beat confectioners' sugar and 3/4 cup butter together in a bowl using an electric mixer until smooth; add melted chocolate and 2 teaspoons vanilla extract and mix until frosting is smooth. Spread frosting over cooled cookie cake.
