Warm the cream and milk together in a saucepan until steaming and keep them beside the stove. Set a bowl in an ice bath with a fine sieve resting in it, and put the chopped chocolate in a separate heatproof bowl.

Stir the 3/4 cup sugar and the water together in a heavy saucepan until the sugar is wet, then bring to a boil and do not stir again. Cook 8 to 10 minutes to a deep amber, about 350F to 355F, swirling the pan once or twice near the end. Take it off the heat the moment it smells nutty.

Pour the hot cream and milk into the caramel a little at a time, standing back because it will seize and spit. Return the pan to low heat and stir until every hardened piece has dissolved, 2 to 3 minutes.

Whisk in the cocoa powder until it is completely smooth. Doing this now, while the liquid is hot and thin, is what stops it clumping later.

Whisk the yolks with the 1/4 cup sugar and the fine sea salt until pale. Pour about half the hot chocolate cream into them in a thin stream, whisking constantly, then return everything to the pan.

Cook over low heat, stirring with a flat spatula and scraping the corners, until it reaches 175F to 180F and coats the back of the spoon, 4 to 6 minutes. Do not let it boil.

Pour the hot custard over the chopped chocolate and whisk until smooth, then strain it through the sieve into the bowl in the ice bath. Stir occasionally until cold, then cover and refrigerate at least 4 hours, preferably overnight.

Churn 20 to 25 minutes in a machine whose bowl has been frozen a full day. Fold the coarse smoked salt through straight off the paddle, pack into a shallow container with parchment pressed onto the surface, and freeze 4 hours before serving.