Rinse the adzuki beans, then heat them in a pot with plenty of water over high heat. Bring to a boil, then stew for 5-6 minutes over medium heat.
Drain in a colander to remove the scum.
Return the adzuki beans to the pot, fill it with plenty of water, then bring to a boil over high heat. Reduce to low heat and simmer for 40-60 minutes.
Once the liquid boils down, add enough water to keep the adzuki beans covered. Turn off the heat once the beans are soft enough to mash between your fingers.
Drain out the liquid, mix in the sugar, cover with a lid and let sit for 1-1 and 1/2 hours.
Return to heat and simmer while stirring from the bottom of the pot with a wooden spatula. When the beans become thick, reduce to low heat, stir in the mizuame, and add the salt.
Transfer to a bowl after it feels cool to the touch. Use the leftover liquid from Step 5 to add sweetness to other dishes.
To store the tsubu-an, cool it, wrap individual portions in plastic wrap, then place in a ziplock bag and freeze. Press out the air before freezing.
Story Behind this Recipe
I learned this method from my mother. Covering the pot and stewing the beans on medium heat speeds up the process. Keep an eye on the pot. In Step 2, if the scum doesn't bother you, skip the steps for draining and refilling the pot with water. For 3 types of botamochi and ohagi, see Recipe ID: 623807.
Once the tsubu-an cools it will stiffen, so turn off the heat while it is still on the soft side. If you don't have any mizuame, it's fine without it. If you're in a hurry, skip Step 4 and reheat right away. Adjust the amount of sugar to taste.