If using roasted sesame seeds, put them in a blender or grind them until they're sticky (see Hints). Regular ground sesame seeds are fine too.
Dissolve the miso in 1 teaspoon of sake (not listed).
Add the ingredients in the listed order starting from the top (excluding the milk), and mix together.
When you add the vinegar it'll become a bit crumbly, but add the sake -> soy sauce and mix well. It'll become smooth.
Once it's as smooth as this, it's done . Finally, add the milk and adjust the thickness. If you like it thick, then omit the milk! Be careful not to make it too thin.
Once it's mixed with the dashi, it'll have a thick texture, but if you think it's too thick, then thin with dashi in the pot, or adjust with milk in Step 5.
Story Behind this Recipe
Instead of just replicating the store-bought flavor, I modified it to our family's taste.
This tastes best when made with sesame seeds that have been roasted to the point where they're exuding oil, but you can make this with unroasted ground sesame seeds too. A food processor is handy for this recipe, as well as commercial sesame paste. You can thin the sauce with milk, which makes it mild and delicious. However, if you make it too thin it'll too bland when you combine it with the dashi stock in the pot, so watch out for that.