Special equipment: An ice cream maker
Heat 1 cup of the sugar in a dry, heavy 10-inch skillet over medium heat, stirring with a fork to heat the sugar evenly, until it starts to melt. Stop stirring and cook until it is dark amber, swirling the skillet occasionally so the sugar melts evenly.
Add 1 1/4 cups of the cream (the mixture will spatter) and cook, stirring, until all of the caramel has dissolved. Transfer to a bowl and stir in the sea salt and vanilla. Cool to room temperature.
Meanwhile, bring the milk, remaining 1 cup cream and remaining 1/4 cup sugar just to a boil in a small heavy saucepan, stirring occasionally.
Lightly whisk the eggs in a medium bowl, then add half of the hot milk mixture in a slow stream, whisking constantly to temper. Pour back into the saucepan and cook over medium heat, stirring constantly with a wooden spoon, until the custard coats the back of a spoon and registers 170 degrees F on an instant-read thermometer (do not let it boil). Pour the custard through a fine-mesh sieve into a large bowl, then stir in the cooled caramel.
Chill the custard, stirring occasionally, until very cold, 3 to 6 hours. Freeze the custard in an ice cream maker according to the manufacturer's instructions (it will still be quite soft), then transfer to an airtight container and put in the freezer to firm up.