Heat a large nonstick sauté pan over medium heat (I use a 12-inch pan). Use a nonstick pan or else things can get messy (see note 7). Add the walnut oil and swirl to coat the pan with the oil. Add the pureed beans and cook for 2 minutes, stirring frequently.
Add the sugar to the pan and stir to combine with the beans. If you're using kansui, add it now, too. Reduce the heat to medium-low and continue cooking the paste for another 8 to 10 minutes. Eventually, the paste will thicken so much that you can fold up the paste with a flat spatula. The consistency of the paste will almost be like soft play-doh.
To check the sweetness of the bean paste, take a pea-sized portion of the paste and let it cool for 30 seconds before tasting. Add more sugar, if desired.
Turn off the heat and add the coconut oil to the pan. Use a flat spatula to fold the oil into the paste. Continue folding until the oil is fully absorbed into the paste.
Transfer the paste to a mixing bowl and let cool for 20 minutes before using. The red bean paste crusts easily, so cover the bowl with a damp towel.
Refrigerate any unused red bean paste for up to a week. You can also freeze the paste in a freezer bag and use it within a few months. Make sure to remove as much air from the bag as possible before freezing.