Heat 1 cup sugar in a dry 10-inch heavy skillet over medium heat, stirring with a fork to heat sugar evenly, until it starts to melt, then stop stirring and cook, swirling skillet occasionally so sugar melts evenly, until it is dark amber.

Add 1 ¼ cups cream (mixture will spatter) and cook, stirring, until all of caramel has dissolved. Transfer to a bowl and stir in sea salt and vanilla. Cool to room temperature.

Meanwhile, bring milk, remaining cup cream, and remaining ¼ cup sugar just to a boil in a small heavy saucepan, stirring occasionally.

Lightly whisk eggs in a medium bowl, then add half of hot milk mixture in a slow stream, whisking constantly. Pour back into saucepan and cook over medium heat, stirring constantly with a wooden spoon, until custard coats back of spoon and registers 170°F on an instant-read thermometer (do not let boil). Pour custard through a fine-mesh sieve into a large bowl, then stir in cooled caramel.

Chill custard, stirring occasionally, until very cold, 3 to 6 hours. Freeze custard in ice cream maker (it will still be quite soft), then transfer to an airtight container and put in freezer to firm up.