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Put the sugar, eggs, and egg yolks in a medium-sized bowl, or in a stand mixer. Beat on medium speed for about two minutes, until the mixture is pale yellow, smooth and thick.
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Put the milk and cream in a medium pan on the stove. Barely bring the mixture to a boil, then remove from heat.
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Pour one cup of the hot milk and cream into a liquid measuring cup. Turn the mixer back on to medium-low speed for the eggs and sugar, and slowly pour the hot milk/cream mixture into the eggs. When thoroughly mixed, pour that entire mixture into the rest of the milk/cream mixture on the stove. Turn the heat on to medium-low, and stir constantly until mixture thickens and can coat the back of a spoon.
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Transfer to a bowl, cover with plastic wrap placed directly on the custard, and chill completely for several hours.
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When completely chilled, remove from refrigerator and add to your ice cream maker and follow the manufacturer's directions. Transfer to an airtight container and freeze for several hours.