Stir together the half-and-half and 2 cups sugar in a medium saucepan over medium-low heat. Split the vanilla bean, scrape out the caviar, and add it to the mixture (or add the vanilla extract.) Heat the mixture until it's hot but not simmering or boiling.
In a large bowl, beat the egg yolks vigorously with a whisk until they start to lighten in color, about 2 minutes.
Next, grab a ladle of the hot half-and-half mixture and very slowly drizzle it into the bowl with the egg yolks, whisking the whole time. (The purpose of this is to temper the egg yolks and bring them to a good temperature before adding them to the pan.) Repeat with a second ladle of hot half-and-half mixture, making sure to whisk the whole time.
Pour the tempered yolks into the saucepan with the rest of the half and half mixture and stir gently with a wooden spoon for 2 to 4 minutes, cooking it slowly until it's thick enough to coat the spoon.
Pour the thick liquid through a fine mesh strainer and into a clean bowl. Add the heavy cream to the bowl and stir gently to combine. Refrigerate this mixture for 2 hours, or until chilled.
Meanwhile, combine the strawberries and 2 tablespoons of sugar in a blender. Puree until smooth (or you can stop just short of smooth if you want a little texture.) Pour the pureed strawberries into the chilled custard mixture and stir.
Now pour this mixture into your ice cream maker and freeze it according to its directions. (You may have to do it in two batches depending on its size.) After it freezes, transfer it to a freezer-safe container and freeze it for at least 4 hours, or until frozen firm.
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