Pulse hazelnuts with a quarter cup of the sugar and salt in a food processor until finely chopped. Transfer to a 3-quart saucepan, add the basil leaves, milk and cream, and bring just to a simmer over moderate heat, stirring occasionally. Remove from heat and let steep, covered, for one hour.

Pour milk mixture through a fine-mesh sieve into a bowl, pressing hard on solids; discard solids.

Use an electric mixer to beat together yolks and remaining six tablespoons sugar in a bowl at medium speed until thick and pale, two to three minutes. Beat in milk mixture and pour into cleaned saucepan. Cook over moderately low heat, stirring constantly with a wooden spoon, until custard is thick enough to coat back of spoon and thermometer registers 175°F; do not let boil.

Pour custard through cleaned sieve into a metal bowl. Refrigerate, uncovered, stirring occasionally, until cool, about one hour, then cover and refrigerate until very cold, three to six hours. Freeze custard in an ice cream maker. Transfer gelato to an airtight container and put in freezer to harden for about one hour.