Bring milk to a simmer and remove from heat. Add tea and allow to steep for 30 minutes. Strain out tea.
Preheat oven to 350°F. Grease and flour three 6-inch round cake pans. Tap out any excess flour. Line with parchment paper and set aside.
In the bowl of a stand mixer, whisk together 2 cups flour (reserve remaining ½ cup), sugar, baking powder, and sat. Using a paddle attachment, run mixer on low speed. With the mixing running, add butter in three additions. Continue to mix on low speed until mixture is crumbly.
In another bowl, combined infused milk, egg, egg white, and vanilla. Add half of liquid to mixing bowl. Mix on medium speed until incorporated. Add remaining half of liquid and mix until batter is thoroughly combined. Scrape down bowl as needed to ensure thorough mixing. The batter may look curdled. Add remaining 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 the cake pan to loosen cake. Invert cake onto a wire rack and allow to cool to room temperature.