Make the beurre manie by mixing and kneading together the softened butter with flour. Set it to the side. Next, bring the milk and 1/3 cup of sugar to a boil in a medium-sized sauce pot.
Turn the heat to low and whisk in the beurre manie until no chunks remain and it is smooth. Boost the heat back to high and bring the mixture to a boil while constantly whisking.
The mixture will become very thick. Next, turn the heat back down to low and occasionally whisk for 5 to 6 minutes or until the starchy flour taste is gone.
Add the chocolate to a metal bowl and place over a medium-sized sauce pot 1/3 of the way filled with water simmering over low to medium heat and mix with a rubber spatula until melted.