Bring chestnuts, milk, and salt to a simmer in a medium saucepan over medium heat. Partially cover; cook until chestnuts are very soft and milk is reduced to 1/4 cup, about 15 minutes. Process mixture in a food processor until very smooth (if needed, add more milk, 1 tablespoon at a time).
Pass through a fine sieve into a bowl; discard solids. Press plastic wrap directly onto surface. Refrigerate puree 1 hour.
Stir together sugar and 1/3 cup water in a small saucepan until sugar is dissolved. Bring to a boil over medium-high heat; cook until syrup registers 238 degrees on a candy thermometer (carefully tilt pan to fully submerge thermometer tip).
Meanwhile, put yolks in the bowl of an electric mixer fitted with the whisk attachment. With mixer running on low speed, gradually pour syrup down side of bowl. Increase speed to high; beat until pale, fluffy, and cool, about 8 minutes.
Whisk cream in a clean mixing bowl until soft peaks form; set aside.
Stir egg mixture into puree; whisk until smooth. Gently fold in whipped cream. Chill, with plastic wrap directly on surface, at least 2 hours or overnight.