Heat 1 cup sugar in a heavy skillet over medium heat stirring to heat sugar evenly. When it’s all melted, stop stirring. Cook, swirling skillet occasionally so sugar cooks evenly until it is dark amber. The darker, the more flavor, but don’t let it burn.
Add 1 ¼ cups cream (careful – it spatters) return to heat and cook, stirring until all the caramel has dissolved. Stir in sea salt and vanilla. Let cool.
Bring milk, remaining cup cream, and remaining ¼ cup sugar to a boil, stirring occasionally.
Whisk eggs in a bowl, then temper with half the hot milk mixture in a slow stream, whisking constantly. Return to saucepan and cook over medium heat, stirring constantly with a wooden spoon, until custard coats back of spoon and registers 170°F. Now that the custard includes eggs, do not let it boil. Pour custard through a fine-mesh sieve to remove any cooked egg solids, then stir in cooled caramel.
Chill custard 3 to 6 hours, stirring occasionally. Freeze custard in ice cream maker. When it firms up to your liking, eat it! If there’s any left, store in the freezer.