In a large mixing bowl, add softened butter. Beat with an electric hand or stand mixer until smooth. Add the vegetable oil and combine.
Pour in the granulated sugar, powdered sugar, egg, and vanilla extract. Beat together until smooth.
Add the flour, baking soda and salt as well as the contents of the chai tea bag or the chai tea spices. Fold everything together until combined. Cover the bowl with plastic wrap and store in the fridge for about an hour.
Once chilled, Preheat oven to 350°F. Grease a baking sheet or line it with parchment paper. Set aside. Scoop out even, tablespoon-sized drops of cookie dough. Roll into balls, cover each one in cinnamon sugar, then very gently press the cookies down slightly with the bottom of a glass or flat surface.
Bake for 8-10 minutes in the oven, or until the sides turn slightly golden. Remove the cookies from the oven and let them cool on the sheet for 5 minutes, then transfer them to a cooling rack to cool completely.
While cooling, create an icing made with 1 1/2 cups of powdered sugar and 3 teaspoons of eggnog. Mix together until smooth.
Once cooled completely, spread a spoonful of icing onto the top of the cookies. Let that harden, then enjoy!