Combine the milk, chopped chocolate, sugar, and chile de arbol in a medium saucepan and bring to a simmer over low heat, stirring until the chocolate melts and the sugar dissolves. Add the coffee, remove from the heat, and let cool. Cover and refrigerate until cold, at least 2 hours and up to 1 day.
Remove the chile before serving. Serve in tall glasses over ice. Grate chocolate on top, if desired.