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Beat eggs together until a uniform color.
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Combine all ingredients except vanilla in a heavy pan (stove) or bowl (microwave).
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Stir well.
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Cook over medium-low heat, stirring constantly (stove). If using microwave, cook for 2 minutes on high, stir thoroughly (to make sure you don't end up with tiny bits of cooked egg), 2 more minutes, stir, etc.
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Cook until thick enough to coat the back of a spoon, at least 160°.
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Allow to cool to about room temperature or, if making ahead, refrigerate until ready to freeze.
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If it looks a little lumpy from the eggs, strain.
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When ready to freeze (make in the ice cream maker), stir in vanilla.
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Pour into the canister of an ice cream freezer and follow its directions.
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When done, if it's too soft to eat, completely cover canister with ice while still in the ice cream maker (depending on your ice cream maker) or put in the freezer until hard enough to eat.