In the bowl of a stand mixer fitted with the paddle attachment, cream together the butter, vegetable oil, 1/2 cup sugar and powdered sugar for 2 minutes. Add the egg and vanilla extract and beat just until combined.
In a separate medium bowl, whisk together the flour, baking soda, chai tea spices and 1/4 teaspoon kosher salt. Add the flour mixture to the stand mixer and beat just until combined.
Cover the bowl with plastic wrap and chill for 1 hour or up to overnight.
When ready to bake, preheat the oven to 350°F. Line two baking sheets with parchment paper.
Scoop out heaping 1-tablespoon portions of the dough and roll them into balls.
Add the remaining 1/4 cup sugar to a small bowl then coat the dough balls in the sugar.
Add the dough balls to the lined baking sheets, spacing them at least 2 inches apart.
Bake the cookies until they are set, 9 to 12 minutes, then remove them from the oven and let them cool completely on the baking sheets while you make the icing.