In a heavy saucepan, gently heat the sugar and water over medium heat without stirring. Let it melt and slowly turn a deep amber color. Once the color develops, carefully whisk in the warm cream, followed by the butter and sea salt. Set the caramel aside to cool slightly.

In another saucepan, warm the milk with sugar until just steaming. In a bowl, whisk the egg yolks, then slowly pour the warm milk into the yolks while whisking constantly. Return the mixture to the pan and cook gently, stirring until it thickens enough to coat the back of a spoon.

Remove from heat, stir in the vanilla, and strain if needed. Combine the custard with the caramel, mix gently, and chill completely.

Churn according to your ice cream maker instructions, then freeze until softly set.