Preheat the oven to 350F; Grease and line pans or cupcake tins, set aside.
Over a low heat, warm the milk to a simmer and stir in the tea leaves. Bring back to a simmer and then turn off and set aside to cool and steep for at least 15 minutes.
In a mixing bowl, beat butter and sugar together until light and fluffy. Add the eggs one at a time and combine. Beat in the oil and vanilla. Gradually incorporate the flour, baking powder, baking soda and salt into the mixture. Strain the tea leaves out of the milk. Stir in the milk, mixing until well combined.
Transfer batter to prepared pans, filling half way and bake for 25 to 30 minutes, or until risen, golden and a toothpick when inserted comes out clean. Remove from the oven, allow to cool before icing.