Brown the butter: melt butter in a large sauté pan over medium heat, then once melted, continuously whisk until brown bits start to form at the bottom of the pan and the butter is foaming, golden in color, and has a nutty aroma. This will take about 1-5 minutes (depending on butter and pan used). Remove from the heat, quickly whisk in the tea leaves, then carefully pour into a small glass bowl to stop the cooking. Cool slightly before proceeding with the recipe.
In a large bowl, whisk together brown butter, oil, granulated sugar, powdered sugar, egg, and vanilla until smooth.
Whisk in the flour, baking soda, and salt until combined, then cover and chill for 30-60 minutes in the fridge.
Line 2 rimmed baking sheets with silicone baking mats and pour granulated sugar (maybe 1/3 cup) into a small bowl. Set aside.
Use a cookie scoop to scoop the dough (you’ll be making 15 cookies) onto the prepared baking sheets, roll each cookie in the sugar, then place back onto the cookie sheets.
Place the cookie sheets in the fridge to chill while you preheat the oven to 350 degrees F.
Baking one sheet at a time, bake cookies in preheated oven for 10-13 minutes, or until very lightly browned on the edges.
Cool on baking sheet for 2 minutes, then transfer to a wire rack to continue cooling before adding optional glaze.
Repeat with remaining cookies.
Glaze
Whisk together powdered sugar, ginger, cinnamon, all-spice, nutmeg, salt, and cloves, then whisk in just enough milk to make a drizzle-able glaze.
Spoon over cooled cookies, then (if desired) allow to harden for about an hour at room temperature before enjoying!