For the azuki paste: Put beans in a saucepan full of water, bring to a boil, then drain in a colander. Return beans to pan, add 3-4 cups water, cover, and simmer until beans are very soft and have absorbed all the water, 30-40 minutes. (Azukis cook faster than many other varieties of bean.) Stir in sugar and cook over low heat until sugar has dissolved and beans form a soft, chunky paste. Add salt and allow paste to cool.
For the ice cream: Slice vanilla beans lengthwise down the center with a sharp paring knife. With your fingers, spread beans open, then scrape out and reserve black seeds, discarding pods. Put seeds in a small bowl, add ¼ cup of the sugar, and mix well.
In a medium bowl, combine remaining ¾ cup sugar, vanilla–sugar combination, and egg yolks, and beat until mixture becomes creamy and turns a pale yellow. Add milk and cream and mix well into a custard-like consistency. Refrigerate until ready to use, preferably overnight.
Pour custard into an ice cream maker and process according to manufacturer's instructions. About 10 minutes before ice cream is ready, feed ½ cup of the azuki paste into ice cream maker. Serve immediately or freeze in an airtight container to keep ice cream soft until ready to use.