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