Whisk flour, baking soda, salt, and dry tea from the tea bag in a small bowl. Set aside.
Beat butter, vegetable oil, and granulated sugar until smooth, then mix in powdered sugar.
Mix in egg and vanilla, then add the dry ingredients to the wet ingredients and mix until smooth. Dough will be wet.
Use a 1-tablespoon cookie scoop to scoop balls of dough on a cookie sheet lined with parchment paper. If your dough is too loose to scoop you can chill it 30 minutes, but mine was okay to scoop.
Chill cookie dough balls for 1 hour.
Preheat oven to 350°F and line 2 cookie sheets with parchment paper or silicone baking mats.
Stir cinnamon and sugar in a small bowl. Roll each cookie dough ball in cinnamon sugar and place 2-inches apart on cookie sheets. Press a bit flat with the palm of your hand.
Bake 7-9 minutes or until flattened and light golden around the edges and on the bottom.
Cool completely before frosting.
Make the frosting by whisking powdered sugar, cinnamon, and nutmeg in a medium bowl.
Add 2 tablespoons milk and whisk, adding more milk or cream until desired consistency (see note).
Frosting will dry a bit but will still transfer, only stack between sheets of parchment or wax paper. Store in an airtight container for several days or freeze for up to 2 months.