In a medium mixing bowl, add in the egg yolks and whisk to combine. Set aside. Save the egg whites for another recipe.
In a medium saucepan, add in the heavy cream, milk, sugar and tea bags. Place over medium heat and let come up to a simmer. Let simmer for 3 minutes. Stir occasionally.
Remove tea bags from the cream mixture. Give it a squeeze to get all the cream and flavor out of the tea bag first.
Add about ½ cup of the hot cream into the egg yolks and whisk constantly until incorporated.
Add about another ½ cup of the hot cream into the egg yolks and whisk constantly until incorporated. This is tempering the egg yolks so they don't cook in the hot cream.
Pour the yolk mixture into the saucepan with the remaining cream mixture and whisk. Let it heat back up and thicken. Once it is thick enough to cover the back of a wooden spoon, it's done. It took about 4 minutes for me
Strain the mixture with a fine mesh strainer. Stir the vanilla extract into the ice cream.
Place the hot mixture into a bowl, cover with plastic wrap and let it chill overnight.
Place mixture in the ice cream maker for the recommended time required by your ice cream maker. Mine recommends no longer than 20 minutes.
Pour the ice cream into a freezer-safe bowl or loaf pan. Cover with plastic wrap and place into the freezer for 2 hours or until frozen.