Put half-and-half, heavy cream, and 1/2 cup sugar into a saucepan, then scrape the seeds from the split vanilla bean into liquid and then add the vanilla bean to the mixture. Gently warm over low flame until the sugar has dissolved. You may want to use a heat diffuser or a double boiler to make the custard unless you are very comfortable with your stove’s settings.
Place the egg yolks and the remaining 1/2 cup of sugar in a bowl and beat until the mixture thickens and turns a pale yellow.
Add about 1/2 cup of the warm cream mixture to the egg yolk mixture and heat until blended. Gradually add the rest of the warm cream mixture to the egg yolks, beating continually. Return the custard to the saucepan and cook gently until it thickens enough to coat the back of the spoon. (Note: Don’t rush this process as the custard will curdle. Stir constantly with a whisk as the custard thickens.)
Remove the vanilla bean from mixture then strain mixture through a fine mesh sieve. Scrape the remaining seeds from vanilla bean into the custard mixture and stir until well blended then add back into the blend. Add the extract to the custard.
Allow custard to chill thoroughly (ideally overnight so that it develops a really deep flavor). Make the ice cream according to the directions of your ice cream maker.
After making the ice cream, pack it into a container and allow it to cure for an hour or so before eating. However, if you love soft-serve ice cream, feel free to dive in!