Combine the milk, cream, sugar and popcorn in a large saucepan set over medium heat. Cook until edges begin to bubble. Cover, remove from heat, and allow to steep for 30 minutes at room temperature.

Strain the milk mixture through a fine mesh strainer, pressing on the popcorn to remove as much liquid as possible.

Whisk together the egg and vanilla in a medium bowl. Slowly pour the warm liquid over the egg mixture, whisking constantly, then pour the mixture back into the saucepan. Continue cooking over medium heat, stirring constantly and scraping the bottom, until the mixture is thick enough to coat the back of a spoon.

Strain and cool in an ice bath, then chill completely in the refrigerator. Freeze the mixture in an ice cream maker according to manufacturer’s directions.

While the ice cream is churning, make the caramel. In a large, deep saucepan, spread the sugar in an even layer and place over low to medium low heat. Watch it carefully. Once it starts to liquefy around the edges, use a rubber spatula to very gently stir it towards the center. Continue gently stirring until all of the sugar is melted, but take care not to over stir.

Once the caramel reaches a deep amber color, immediately remove it from the heat. Carefully whisk in half the cream, which will bubble and steam quite violently. Stir until the cream is thoroughly combined, then whisk in remaining cream. Stir in the butter and salt. If any sugar has hardened, place the saucepan back over low heat and whisk until smooth.

The caramel sauce can be stored, refrigerated, for one month. To rewarm, simply microwave or place in a saucepan over low heat.