Make the sauce: Heat 1 cup of sugar in a large heavy skillet over medium heat. Stir until the sugar starts to melt; then cook, stirring occasionally until the melted sugar turns a dark amber color. Add 1 1/4 cups cream (be warned, it will splatter); continue to cook, stirring until all the caramel has dissolved. Remove from heat and pour caramel sauce into a bowl. Stir in 1/2 teaspoon of sea salt and 1/2 teaspoon of vanilla. Set aside and let cool to room temperature.
Make the ice cream base: Whisk the egg yolks in a medium-size heat-proof bowl until smooth. Gradually whisk in the sugar until the mixture is thick and pale yellow. Set aside. Combine the milk and cream in a heavy-bottomed saucepan and cook over medium heat, stirring occasionally, until an instant read thermometer reads 170 F. Remove mixture from heat. To temper the egg yolks, pour a splash of the hot milk mixture into the yolk mixture and whisk rapidly. Then pour all of the yolks into the milk and cream and return the mixture to the heat. Cook, stirring frequently, until the custard base thickens and the instant read thermometer reaches 185 F. Remove the custard base from the heat and pour through a fine mesh strainer into a clean bowl. Let strained mixture cool for 5 minutes. Then mix in the reserved caramel sauce. Cover caramel-cream base and chill in the refrigerator until completely cool.
Pour the chilled custard base into an ice cream maker and process according to the manufacturer's instructions. Transfer ice cream to an air-tight container and freeze for at least 2 hours before serving.