Preheat the oven to 350 F and line two baking sheets with parchment paper.
Beat the butter, brown sugar, and granulated sugar until light and creamy. Mix in the egg and vanilla until smooth.
Whisk the flour, cinnamon, ginger, cardamom, cloves, nutmeg, baking soda, and salt in a separate bowl. Add the dry ingredients to the butter mixture and mix just until combined.
Divide the dough into 12 portions and arrange them at least 2 inches apart on the prepared baking sheets. Gently flatten the tops.
Bake for 10 to 12 minutes, until the edges are set and the centers remain soft. Cool the cookies completely on a cooling rack.
Whisk the powdered sugar, black tea, heavy cream, vanilla, and cinnamon until smooth. Spoon the chai glaze over the cookies and let it set for 20 minutes.