Bring milk to a simmer. Add tea and steep for 30 minutes. Strain out tea. Squeeze out as much liquids from tea as possible. Measure remaining liquid. Add additional milk as necessary until you have ¾ cup liquids.
Preheat oven to 350° F. Lightly butter and flour three 6-inch round cake pans. Line with parchment paper. Set aside.
In the bowl of a stand mixer fitted with a paddle attachment, combine 2 cups flour (reserve remaining ½ cup), sugar, baking powder, and salt. Run mixer on speed. Add butter in three additions. Continue to mix on low speed until mixture is crumbly.
In another bowl, combine infused tea milk, eggs, and egg white. Add half of liquid to mixing bowl. Mix on medium speed until incorporated. Scrape down bowl as needed to ensure thorough mixing. Add remaining half of liquid and mix until batter is combined. The batter will look curdled. Add reserved ½ cup flour and mix until batter comes together.
Evenly distribute batter among prepared cake pans. Bake for 30-35 minutes until toothpick inserted in center of cake comes out clean. Allow cake to cool in pan for 10-15 minutes. Run an offset spatula along the sides of cake pan to loosen cake. Invert cake onto a wire rack and allow to cool to room temperature.