Using a stand mixer, beat butter until creamed. Add the oil and mix on medium for about thirty seconds. Scrape sides and bottom of the bowl and add the granulated sugar, powdered sugar, egg, vanilla, and the contents of the chai tea packet. Mix on medium until everything is incorporated.
On low, mix in flour, baking soda, and salt until just combined.
Refrigerate dough for about 1 hour until firm enough to handle.
Preheat your oven to 350 degrees F. Line 3 baking sheets with parchment paper or grease well.
Scoop heaping tablespoons of dough (just over 1 ounce) onto your cookie sheet, 8 to a sheet (these cookies spread!). With the palm of your hand, flatten dough until it is 1/4-inch thick. For more uniform-looking cookies, roll your dough into balls before you flatten them.
Bake for 8-10 minutes until the edges are just lightly golden. Cool completely before adding the glaze.