In a small pot, bring the milk, 1 cup of the cream, honey, vanilla, and pinch of salt to a simmer. Stir enough to dissolve the honey. Remove from heat, and set the tea bags into the hot cream. Steep, uncovered, for 15 mins. Taste the cream mixture before removing the tea bags. You should want it to taste like a strong chai latte. You can leave the tea bags in for as little or as long as you'd like - it simply depends on how strong you want the chai flavor to be. Keep in mind that you're going to add in another cup of cream later, and that will dilute the flavour quite a bit.
In a large bowl, whisk together the egg yolks. As you whisk, slowly pour a small amount (approx. half of a cup) of the hot cream mixture into the egg yolks to temper them. Now transfer the yolk mixture into the cream mixture. Heat the mixture over low heat, stirring occasionally, until the mixture is thick enough to coat the back of a spoon. Stir in the other cup of cream.
Let the mixture stand to cool, and once lukewarm in temperature, refrigerate for at least 2 hours, if not longer.
Churn in your ice cream maker according to manufacturer's instructions. Five minutes before the ice cream is done, slowly add the crumbled cookies.