Combine the cream, milk, and sugar in a medium, heavy saucepan. Combine the flour and the water and whisk into the mixture. 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.
With a food processor, chop the berries until fine. Set aside. Turn on the ice cream maker and pour in custard. When the ice cream is 3/4 ready, pour in the berries and finish freezing. Serve immediately.