Gather all of your ingredients and have them nearby ready to add to the mixture as needed. In a heavy-bottomed 3 quart saucepan, heat sugar, corn syrup and water over moderately high heat whisking constantly to dissolve sugar.
Once mixture reaches a boil, stop whisking and allow mixture to boil until it reaches a dark amber color, carefully swirling pan occasionally (be careful not to burn it, it should become a deep orange with a bit of red/brownish color).
Once mixture reaches a dark amber color, immediately add butter and whisk until butter has melted then immediately remove from heat. Wait 3 seconds then carefully pour in 2/3 cup of the heavy cream and immediately whisk to combine (it will bubble vigorously so be careful).
Whisk until mixture is smooth then stir in vanilla. Set aside to cool slightly while preparing ice cream mixture.
To prepare ice cream:
In a separate medium saucepan, bring remaining 1 1/3 cups heavy cream and 1 3/4 cup of the milk just to a boil over medium-high heat, stirring occasionally. Meanwhile in a small mixing bowl whisk together cornstarch and remaining 1/4 cup milk until smooth.
Once cream and milk mixture begins to bubble, stir in cornstarch mixture and reduce heat to medium-low, then cook stirring constantly until mixture has thickened well and coats the back of a wooden spoon, about 4 - 5 minutes.
Remove from heat and stir in caramel sauce. Transfer mixture to a heat proof container and chill in refrigerator until completely cool, then transfer to freezer and chill until very cold, about 1 - 1 1/2 hours, stirring occasionally during chilling process.
Process mixture in an ice cream maker according to manufactures directions, adding sea salt during the last 5 minutes of mixing. Return ice cream to container, cover and transfer to freezer and freeze until semi-firm or firm.