Combine 1 cup of the cream, the sugar and salt in a medium saucepot and warm over medium heat until the sugar is dissolved. Stir in the remaining cream and the milk and add the tea bags. Simmer over low heat, stirring occasionally, until the tea changes the mixture to a light tan color. Depending on how strong you like your tea, you may want to soak the bags in the mixture overnight in the refrigerator. Chill the mixture thoroughly before churning.
Transfer into a freezer-safe container. Microwave your desired amount of cookie butter until it is warm and pourable, about 20 seconds, and then swirl into the ice cream using a spatula. Freeze before serving.
This recipe was provided by a chef, restaurant or culinary professional and may have been scaled down from a bulk recipe. The Food Network Kitchens chefs have not tested this recipe, in the proportions indicated, and therefore, we cannot make any representation as to the results.