2 cups semisweet chocolate chips
1 (12 ounce) package milk chocolate chips
3/4 cup white sugar
3/4 cup instant hot chocolate mix
3 tablespoons butter
2 (12 fluid ounce) cans evaporated milk
2 teaspoons vanilla extract
4 cups ice
1 cup cold milk
Place semisweet chocolate chips, milk chocolate chips, sugar, hot chocolate mix, and butter into the top of a double boiler over simmering water. Stir frequently, scraping down the sides with a rubber spatula to avoid scorching, until chocolate chips start to melt, about 5 minutes.
Stir evaporated milk slowly into the chocolate mixture until chocolate chips are halfway melted, about 3 minutes. Remove from heat; continue stirring sauce is until smooth and velvety, about 2 minutes more. Mix in vanilla extract. Let sauce cool to room temperature, at least 20 minutes.
Place 1 cup chocolate sauce, 4 cups ice, and 1 cup milk in a blender. Cover and blend until smooth; pour into 2 individual glasses. Refrigerate the remaining sauce.