Makes 1 cup
Put chocolate chips or pieces in a pyrex measuring cup with milk. Heat in the microwave until quite warm. Add vanilla and mix with a stick blender until smooth. Put back in the microwave until it reaches serving temperature, 180°F or so.
For European-style hot chocolate, add 1/2 tablespoon of cornstarch to the warm mixture when blending. This should be heated until it simmers, removing it from the microwave to stir every 30 seconds or so. You can thicken it more or add some sugar if you like it sweeter.
If you choose a bittersweet chocolate, you may want to add some sugar as well.
Makes 1 quart
Put the milk, chocolate, and vanilla in a glass pitcher in the microwave and heat until quite warm but not hot. Blend with a stick blender until smooth. Put in the refrigerator and chill. When cold, stir before serving.