On medium heat, heat 1 cup sugar in a saucepan, until it melts and gets a dark amber color. Add 1 1/4 cups of heavy cream and stir until all the caramel has dissolved. Transfer to a bowl and stir in sea salt and vanilla. Cool to room temperature.
In the same saucepan, bring milk, remaining cup of heavy cream, and remaining 1/4 cup sugar to a boil, stirring occasionally. Lightly whisk eggs then add half of hot milk mixture in a slow stream, whisking constantly. Pour back into the saucepan and stir constantly with a wooden spoon.
Cook on medium heat until custard coats back of a spoon (don’t let it boil). Pour custard through a fine-mesh sieve into a large bowl. Don’t skip this part, you don’t want scrambled eggs in your ice cream. Stir in cooled caramel then place in the refrigerator for 3 to 6 hours, stirring every half hour.
After 3-6 hours, place the mixture into the ice cream maker. Follow the directions of your ice cream maker. Once the ice cream is done, transfer it to an airtight container and fold in chocolate-covered waffle cone pieces. Place in freezer to firm up.