In a bowl, whisk together the egg yolks, sugar, vanilla, and salt. Pour the half-and-half and cream into the top of a double boiler and heat over hot water until hot, but not boiling. Stir with a wooden spoon, scraping the bottom of the pan to prevent sticking. Slowly pour the hot cream into the eggs and whisk until smooth.
Return to the double boiler and cook over medium-low until thick and bubbling around the edges. The custard will coat a wooden spoon. Strain the custard through a fine-mesh sieve into a metal bowl. Cover and chill for at least 4 hours or overnight.
Rinse and hull the berries. Roughly chop the berries and transfer half to a small bowl. Mash the other half with a fork or potato masher; they will not be smooth but still a little chunky. Add them to the chopped berries. Chill for at least 1 hour. Pour the cream into the canister of an ice cream maker. Freeze according to manufacturer's instructions. When the ice cream is the consistency of whipped cream, stir in the berries. Transfer to a shallow quart container, cover with plastic wrap, and freeze in the refrigerator's freezer for at least 3 hours until ready to serve. Let the ice cream sit at room temperature for about 10 minutes before scooping.