Special equipment: an ice cream maker
Stir together the half-and-half and 2 cups of the 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, cooking it slowly, until it's thick enough to coat the spoon, 2 to 4 minutes.
Set a fine-mesh strainer over a clean bowl. Pour the thick half-and-half liquid through the strainer into the bowl. Add the heavy cream and stir gently to combine. Refrigerate this mixture until chilled, about 2 hours.
Meanwhile, combine the strawberries and remaining 2 tablespoons 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 the manufacturer's directions. Depending on the size of your machine, you may need to do this in 2 batches. Transfer to a freezer-safe container and place in the freezer until frozen firm, at least 4 hours.