In a medium-large bowl, add the vegan butter and granulated sugar. You can use a stand-mixer or hand mixer if preferred. Cream together, then stir in the oat milk and vanilla extract.
In a different bowl, add the flour, cornstarch, baking powder, salt and Earl Grey tea. Stir to combine.
Transfer the dry ingredients to the wet ingredients and combine. Use your hands if needed. The dough will be dry, but it should hold together when pressed. Place a large piece of parchment paper on a clean surface and add a small amount of flour. Place the cookie dough on the parchment paper and roll out the dough to approximately ¼ inch thick. Sprinkle a bit of flour on top (or on the rolling pin) if needed to stop the dough from sticking to the rolling pin. Cover and transfer the dough to the refrigerator to chill for 1 hour.
When you’re ready to bake the cookies, preheat the oven to 350°F. Line 2 large baking sheets with parchment paper (or work in two batches). Remove the cookies from the refrigerator.
Cut the dough into your desired shape. You should get about 24 cookies, depending on the size of your cookie cutter. Gently place the cookies on the prepared baking sheet.
Bake for 8-10 minutes at 350°F.