Pour half-and-half into a heavy saucepan and bring to a boil. Remove from heat. In a bowl, beat egg yolks and 1/2 cup of sugar until pale and smooth. Stir one-quarter of hot half-and-half into egg mixture, stirring well.
Then pour yolk mixture into warm saucepan. Cook over low heat, stirring constantly, until thick enough to coat a spoon, about 3 minutes. Stir in vanilla. Cool to room temperature and cover with plastic. Refrigerate at least 4 hours.
An hour before making the ice cream, in a bowl, combine berries with 2 tablespoons of sugar and let sit for 1 hour. Then mash berries with a fork until they are bite-sized.
Stir into chilled custard mixture and pour into ice-cream maker. Follow manufacturer’s directions. Freeze at least 2 hours before serving.