Eel sauce is called "nitsume" in Japan and is used with eel, shrimp and octopus dishes in Japanese cuisine. In Japanese, "nitsume" actually means "boiling down" and originally was made by simmering eels in a pot of water for several months until a thick glaze was all that was left.
In the U.S., fish stock called dashi is often used instead of eel stock, but nitsume can also be made vegan by using leftover vegetables or vegetable stock when making the sauce. Heat a saucepan and put in 4 tablespoons of dashi or eel stock, 2/3 cup of sugar, 1 cup of mirin and 1/2 cup of sake, into the pan on a high heat and wait for the dashi and sugar to dissolve.
Once it is boiling and the sugar has dissolved, add 1 cup of soy sauce and simmer for 15 to 20 minutes. Mix 1 tablespoon of cornstarch and 2 tablespoons of water. The cornstarch and water makes a slurry, which is used to thicken up sauces without affecting the flavor, but it is optional. If using the cornstarch, pour it in to the mix, stirring continuously. Take the pan off of the heat and keep stirring it until it stops simmering. Once cool, pour it in to a bottle and use it with sushi and fish or eel dishes.