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Make the Cookie Dough: Cream the butter in a mixer fitted with a paddle attachment (or use a hand-held beater) until soft and fluffy.
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Add both sugars and mix.
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Add the vanilla and 1 egg and mix.
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Add the remaining egg and mix.
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Add the flour, baking soda, and salt and mix.
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Add the chips and mix just until just combined.
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Using your hands, roll the dough into a long thin rope; then cut into small bits.
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Refrigerate until ready to use.
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Make the Ice Cream Base: In a saucepan over medium heat, heat the half-and-half, cream, and vanilla, whisking occasionally, to make sure the mixture doesn't burn or stick to the bottom of the pan.
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When the cream mixture reaches a fast simmer (do not let it boil), turn off the heat and let the flavors infuse for 10 minutes.
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Whisk together the egg yolks and sugar.
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In a thin stream, whisk half of the cream mixture into the egg yolk mixture.
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Then pour the egg-cream mixture back into the saucepan containing the rest of the cream mixture.
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Heat over medium heat, stirring constantly with a wooden spoon.
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At 160 degrees F., the mixture will give off a puff of steam.
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When the mixture reaches 180 degrees F, it will be thickened and creamy, like eggnog.
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(If you don't have a thermometer, test it by dipping a wooden spoon into the mixture.
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Run your finger down the back of the spoon.
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If the stripe remains clear, the mixture is ready; if the edges blur, the mixture is not quite thick enough.)
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When it is ready, quickly remove it from the heat.
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Meanwhile, put 2 handfuls of ice cubes in the bottom of a bowl, and add cold water to cover.
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Rest a smaller bowl in the ice water.
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Pour the cream mixture through a fine sieve (to remove the vanilla bean pieces and cinnamon sticks) into the smaller bowl.
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Chill for 3 hours.
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Freeze the cream mixture in an ice cream maker according to the manufacturer's instructions.
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Meanwhile, put a bowl in the freezer.
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When the ice cream is frozen, remove the bowl from the freezer, and transfer the ice cream to it.
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Fold in the chocolate chip cookie dough (you may not need all of it).
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Pack the ice cream into sealable containers and return to the freezer.
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(Make cookies: bake any leftover dough in a preheated 350 degree F. oven for about 10 to 12 minutes.)
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Notes about the recipe: Homemade ice cream is an inexpensive luxury, and an easy one.
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In this recipe, we use lots of fresh cream, not milk, which makes a dense and satisfying mixture.
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The key to great flavor is using whole vanilla beans.
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If you are lucky enough to be choosing among vanilla beans from around the world, here's a quick flavor guide: Mexican beans are creamy, with spicy overtones.
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Madagascar beans are even more creamy, but Indonesian ones have a sweet, woody flavor.
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Tahitian vanilla is very fruity, with an strong flavor of ripe cherries.
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Most extracts are made by blending the four into the basic vanilla flavor we recognize; whole beans have more individuality.
