Preheat oven to 350 degrees. Grease and flour a bundt pan and set aside.
Cream butter in a stand mixer or food processor until smooth and a slightly lighter shade of yellow (about two minutes). Add sugar and continue to cream for three minutes. Then add eggs, one at a time, followed by sweet tea and vanilla extract.
Meanwhile, add flour and sea salt to a large bowl and stir to combine. Add dry mixture to batter in two batches and mix until just combined. Give the batter a few stirs by hand to make sure no dry pockets remain.
Transfer batter into prepared pan and bake on middle oven rack for about 1 hour and 20 minutes or until a toothpick inserted into multiple parts of the cake comes out clean.
Remove cake from oven and let it sit in pan for about 10 minutes then carefully invert it onto a plate to finish cooling to room temperature.
While cake cools, mix up brown sugar and remaining black tea until brown sugar is dissolved. Use toothpick to poke multiple holes into top of cake then spoon glaze over top.