Split the vanilla bean lengthwise. Scrape out the seeds. Put the seeds, pod halves, milk cream and 1/4 cup of the brown sugar in a heavy-bottomed medium saucepan. Bring just to the simmering point over low heat.
In the meantime, combine the remaining 3/4 cup brown sugar, egg yolks and vanilla extract in a large bowl and whisk until just blended.
While gently whisking the yolks, drizzle the hot cream mixture into them so that they are gradually warmed up. Return everything to the saucepan and cook while stirring with a wooden spoon. Make sure that you are constantly scraping the spoon across the bottom of the pan so the custard does not scorch. The custard is done when it has thickened slightly and can evenly coat the back of the spoon. Do not let it come to a boil. Strain the custard through a fine sieve. Press our all the liquid in the vanilla beans and nestle the container holding the custard into a large bowl of ice. Let cool, stirring occasionally.
Transfer the custard to an ice cream machine and freeze according to the manufacturer's instructions. Put the finished ice cream in a storage container and freeze until firm.