Combine 1/4 cup water and the fleur de sel in a small saucepan, set it over medium heat and cook, stirring, just until the salt has completely dissolved, less than 1 minute. Let it cool to room temperature.
Put the ice cream base and the salt mixture into a bowl and whisk to combine. Pour the mixture into an ice cream maker and turn on the machine. Churn just until the mixture has the texture of soft-serve.
Put the caramel in a warm place or warm it in a small saucepan over very low heat just until it’s drizzleable, but not so warm that it’ll melt the ice cream.
Quickly alternate spooning layers of the ice cream and drizzling on a generous spiral of caramel in freezer- friendly containers.
Cover with parchment paper, pressing it to the surface of the ice cream so it adheres, then cover with a lid. It’s okay if the parchment hangs over the rim. Store it in coldest part of your freezer until firm, at least 6 hours. It will keep for up to three months.