Preheat oven to 350°F, and grease a 10- to 12-cup Bundt pan with cooking spray. Set aside.

Make the Cake: Heat milk in a small saucepan until steaming and almost boiling. Remove from heat and add tea bags. Allow to sit for 10 minutes; press milk out of tea bags and discard. Remeasure tea milk and add additional milk until you have exactly 3/4 cup. Set aside to cool completely.

In a medium sized bowl, whisk together flour, baking powder, and salt. Set aside. In the bowl of a stand mixer, cream together butter and sugar for 4 minutes, scraping down the sides of the bowl as needed. Add first three eggs, one at a time, waiting until each is incorporated before adding the next. Add two large spoonfuls of the flour mixture and mix to incorporate. Then add last 3 eggs one at a time.

Mix together tea mixture with vanilla extract and vegetable oil. Alternate adding flour mixture and tea mixture in 3 additions to the mixer, beginning and ending with the flour mixture and mixing for 1 minute with each addition. Scrape the sides of the bowl to ensure everything is well combined. Pour cake batter into prepared Bundt pan and use a spatula to level the batter. Bake until a cake tester inserted comes out clean, 55 to 60 minutes. Allow to cool for 30 minutes before inverting out of pan.

Make Sweet Tea Syrup: Bring water in a medium saucepan to a boil; remove from heat and add tea bags. Allow to steep for 5 minutes. Remove tea bags and discard. Add sugar and salt to tea; heat and stir over medium until sugar is completely dissolved. Set aside to cool.

Use a cake tester to prick the surface of the cake all over. Brush the syrup across the surface of the cake; continue brushing until all the syrup has been absorbed.