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.