Heat the milk in a small pot over medium-high heat, stirring continuously and stopping just before it boils. Steam and small bubbles around the edges signal that the milk is hot enough to melt the chocolate without a rolling boil.

Cover the chopped chocolate with the hot milk and work the mixture with a spoon or whisk until no pieces remain. A glossy, evenly brown surface shows that the chocolate has melted into the milk.

Refrigerate the chocolate mixture for about 4 hours, allowing it to become completely cold. The mixture should feel chilled throughout so it won't immediately melt the ice.

Stir the cold chocolate mixture until uniform, then pour it over ice if choosing the unblended preparation. An even color and consistency show that any settled chocolate has mixed back into the milk.

For the frozen preparation, blend the chilled chocolate mixture with 1 cup ice for about 30 to 45 seconds. Stop when the ice looks finely crushed and the drink circulates as a thick, frosty slush.