Make the Cookie Dough: Beat the softened butter, brown sugar, and granulated sugar for 2–3 minutes, until light and creamy. Mix in the egg and vanilla. In a separate bowl, whisk together the flour, baking soda, salt, cinnamon, ginger, cardamom, cloves, and nutmeg. Stir the dry ingredients into the wet ingredients just until a soft dough forms.
Chill, Steep, and Scoop: Cover and refrigerate the dough for 30 minutes. Steep a chai tea bag for at least 5 minutes to make strongly brewed tea and let it cool completely. Near the end of chilling, preheat the oven to 350°F and line two baking sheets with parchment. Scoop about 3 tablespoons of dough per cookie, roll into balls, and arrange about 2 inches apart.
Bake, Cool, and Make the Glaze: Bake for about 11 minutes, until the edges are set but the centers still look slightly soft. Cool on the baking sheets for 5 minutes, then transfer to a wire rack to cool completely. Whisk the powdered sugar, 2 tablespoons cooled chai tea, vanilla, cinnamon, and cardamom until thick but pourable. Add additional tea ½ teaspoon at a time only if needed.
Glaze: Drizzle over the completely cooled cookies and let the glaze set for 20–30 minutes before serving or stacking.