Combine the half & half and heavy cream in a small saucepan. Split the vanilla bean in half and scrape out the pulp. Add the bean along with all of the pulp to the pan. Heat this mixture over medium-low heat until just simmering (do not boil). While it heats up, whisk the egg yolks vigorously in a mixing bowl until pale yellow. Add the sugar and brown sugar, whisking until incorporated. Temper the egg yolks by slowly adding about 1/3 of the heated cream mixture into the bowl. Once incorporated, pour the tempered egg mixture into the saucepan with the rest of the cream mixture. Cook until the custard thickens and coats the back of a spoon, about 8 minutes over medium-low heat. Transfer the custard to a bowl and stir in the vanilla extract. Let cool slightly before refrigerating and cooling completely. Pour the cooled custard into an ice cream maker and process according to the machine’s instructions. At this point, it will be at the soft-serve stage. I personally like eating it just like this. You can also transfer it to a freezer container and freeze it for a couple of hours before scooping out. (If you don’t have an ice cream maker, check out the writing above.)