Combine the cream, half and half, and 1/2 cup of the sugar in a heavy-bottomed saucepan. Scrape the seeds from half of the vanilla bean using the back of a knife, and add the seeds and that half-bean to the saucepan. Heat on medium-low and cook, stirring constantly, until the sugar is dissolved and bubbles start forming around the edge of the pan. Remove from the heat.
In a medium bowl, whisk the egg yolks, remaining 1/2 cup of sugar, and 1/2 teaspoon of salt until the eggs are pale yellow and fluffy.
Slowly pour about a cup of the hot cream into the egg mixture, while constantly whisking or stirring the eggs.
Add the now-warmed egg mixture to the cream mixture in the saucepan, while constantly stirring or whisking the cream mixture. Turn the heat back on to medium low and cook, stirring or whisking constantly, until the custard is thick enough to coat the back of a spoon — this means that if you dip the spoon in the mixture and run your finger across the spoon, the line will hold.
Prepare an ice bath with a bowl on top of a second bowl that has ice and some cold water in it. Make sure the top bowl can't actually submerge in the water, even after the ice starts melting. Strain the hot custard through a fine-mesh sieve into the top bowl. Scrape the vanilla seeds from the second half of the vanilla pod and add the seeds and bean to the custard mixture.
Let the custard cool, stirring occasionally, until it's no longer warm. Cover the top of the custard with plastic wrap (to avoid the chance of it forming a skin). Refrigerate until fully chilled, about 8 hours.
Remove the vanilla bean and churn the mixture in the ice cream maker according to the manufacturer's instructions.
Transfer to a storage container and freeze until firm.