In a medium pot over medium heat, melt 1 cup sugar with 3 tablespoons water, swirling pot frequently, until sugar turns dark amber in color.
Slowly add heavy cream, milk, remaining ¼ cup sugar, salt and xanthan gum. Simmer the mixture until the caramel melts and the cream is smooth.
While the milk mixture is heating, whisk the egg yolks together until smooth in a heatproof bowl. Temper the egg yolks by very slowly pouring the hot milk mixture while whisking constantly. Pour this mixture back into the saucepan and heat, stirring frequently, on low heat to a temperature of 170°F. The mixture should be slightly thickened, enough to coat the back of a spoon. It should not come to a boil.
Pour the custard mixture through a fine mesh strainer into a clean bowl in an ice bath to cool the mixture completely, stirring constantly. Cover and refrigerate at least 4 hours or overnight.
Remove the custard from the refrigerator and stir in the vanilla extract. Pour the mixture into an ice cream machine and churn for about 20 minutes (follow your ice cream maker directions).
Transfer ice cream to airtight container and freeze until solid, about 2 hours.
Notes
Use the Best Ingredients - when making ice cream you want to use the best ingredients possible so use free-range eggs and organic milk and cream. Also use high quality products when it comes to added flavorings like vanilla or chocolate. Never use imitation vanilla.
Don't Cut the Fat and Calories - salted caramel ice cream is not the dessert that you should be cutting the fat. It depends on the high-fat content to create a creamy texture. If you go lower fat, it will cause an icy ice cream. If you're on a diet, just have a smaller scoop!
Add Extract at Right Time - if you add an extract or alcohol while the custard is too hot, you won't get the actual flavor as it will cook out slightly. Just wait until right before you pour the custard into the ice cream maker to add it.
Chill the Custard - you don't necessarily need to chill the custard overnight, but you want to chill until it is as cold as possible.
Use Frozen Bowl - make sure your ice cream bowl has been frozen for at least 24 hours, otherwise you aren't giving the ice cream a great chance of freeze.
Start Ice Cream Motor First - do not pour the custard in the frozen bowl before starting the ice cream maker. It will start freezing upon contact. You want the motor running and slowly pour the custard in.
Don't Overchurn - always error on the side of caution when it comes to churning your salted caramel ice cream. You do not want it to be hard. Remember it will continue to freeze once you place in the freezer.
Keep It Creamy - homemade ice cream can turn hard as we don't add preservatives, however Xanthan gum or a couple tablespoon of vodka will help keep it creamy.