In a large mixing bowl, cream together the butter, vegetable oil, granulated sugar, powdered sugar, egg, and vanilla extract until smooth and fluffy.
Gently fold in the crushed chai tea leaves until evenly incorporated.
Gradually fold in the flour, baking soda, and salt, scraping the sides of the bowl to fully combine all dry ingredients.
Divide the dough into two equal portions and shape each into a disk. Wrap in plastic wrap and refrigerate for at least 1 hour or until firm.
Preheat the oven to 350°F (175°C) and line baking sheets with parchment paper.
On a lightly floured surface, roll out one dough disk to about ¼ inch thickness. Use cookie cutters to create desired shapes and place them on prepared baking sheets.
Bake for 8–12 minutes, or until the cookie edges turn lightly golden.
Let cookies cool on the baking sheet for a few minutes before transferring to a wire rack to cool completely. Repeat with the remaining dough.
Drizzle cooled cookies with glaze and optionally sprinkle granulated sugar on top for a festive finish.