In a medium saucepan, combine whole milk, 1 cup of heavy cream, honey, vanilla, and salt. Heat over medium heat until it just begins to simmer, stirring to dissolve the honey.
Remove from heat and add the chai tea bags. Let them steep for 30 minutes, then remove and discard the tea bags. Allow the mixture to cool for 15 minutes.
In a separate bowl, whisk the egg yolks. Gradually add about 1/2 cup of the cooled chai-infused mixture to the yolks, whisking constantly to temper them. Pour the tempered yolks back into the saucepan with the remaining chai mixture.
Return the saucepan to low heat and cook, stirring constantly, until the mixture reaches 170–175°F (77–80°C) and thickens enough to coat the back of a spoon.
Pour the custard through a fine mesh strainer into a large bowl to remove any curdled bits. Stir in the remaining 1 cup of heavy cream. Cover and refrigerate until thoroughly chilled, at least 4 hours or overnight.
Once the mixture is cold, churn it in your ice cream maker according to the manufacturer's instructions. Transfer the churned ice cream to an airtight container and freeze until firm, about 2-4 hours.