Whisk flour, white sugar, baking soda, and salt together in a mixing bowl.
Melt butter in a saucepan over low heat; stir in water and cocoa powder until thoroughly combined.
Stir melted butter mixture into flour mixture. Stir in eggs, buttermilk, and 1 teaspoon vanilla extract until the batter is moistened.
Beat batter with an electric mixer on medium speed until thoroughly combined, about 1 minute.
Spoon chocolate cake batter into ice cream cones, leaving about 1 inch of space at the top.
Bake immediately in the preheated oven (to avoid cones becoming soggy) until a toothpick inserted into the middle of a cake center comes out clean, about 30 minutes.
Let cupcakes cool thoroughly, about 1 hour.
Beat cream cheese, confectioners' sugar, and 1/2 teaspoon vanilla extract together in a bowl until smooth and creamy. Frost cupcakes with the cream cheese frosting.