Warm the milk and chocolate pieces together in a pan on the stove over medium heat.
Simmer the chocolate and milk, stirring occasionally, until the mixture begins to boil. When it reaches the boiling point, remove it from the heat. At this point, the mixture won't be very thick yet.
Allow the mixture to cool for around a minute.
Return the chocolate mixture to the stove again, and simmer until it boils again.
Remove the chocolate from the heat source and allow it to cool again.
Repeat the process of simmering until boiling, then cooling, at least one more time or until you are happy with the thickness.
The mixture will continue to thicken as it cools, but it's best served warm, accompanied by churros or sweetbreads, of course.
The process of cycling between boiling and cooling is what thickens the hot chocolate.
If you aren't happy with the thickness, you can further thicken it with cornstarch, tapioca starch, or arrowroot powder. To do so, mix a tablespoon of starch with enough cold milk or water to fully dissolve the starch. Add a small amount of the starch mixture to the warm chocolate, stirring quickly to fully incorporate it before heating the mixture again to boiling point. Add more as needed.
Chocolate a la taza is normally accompanied by churros or sweetbreads.