Pick over the beans and remove any stones or broken beans. Place in a large pot or bowl and cover with water by 3 inches. Soak overnight. Drain and place the beans in a heavy, medium pot with the water. Bring to a boil. Reduce the heat to medium-low and simmer, stirring occasionally, until soft, about 1 to 2 hours. Remove from the heat and let cool.

Strain the beans through a fine strainer set over a medium pot, and discard the pulp. Add the sugar and oil and cook over medium heat, stirring, until the mixture forms a thick, sweet paste, 20 to 30 minutes. Remove from the heat and let cool. Refrigerate before using as a topping for ice cream or in other desserts.

Combine the cream, milk, and sugar in a medium, heavy saucepan. Scrape the seeds from the vanilla bean into the pan and add the vanilla bean halves. Bring to a gentle boil over medium heat. Remove from the heat.

Beat the egg yolks in a medium bowl. Whisk 1 cup of the hot cream into the egg yolks. Gradually add the egg mixture in a slow, steady stream, to the hot cream. Cook over medium-low heat, stirring occasionally, until the mixture thickens enough to coat the back of a spoon and reaches 170 degrees F. on an instant-read thermometer, about 5 minutes. Remove from the heat and strain through a fine mesh strainer into a clean container. Cover with plastic wrap, pressing down against the surface to keep a skin from forming. Chill in the refrigerator for 2 hours.

Remove from the refrigerator and pour into the bowl of an ice cream machine. Freeze according to the manufacturer's instructions. Transfer to an airtight container and freeze until ready to serve.

Spoon the ice cream into bowls and top each with 1 to 2 tablespoons of the red bean paste and garnish with chopped peanuts and crystallized ginger.

Serve immediately.