Reserve 1 tablespoon of brown butter for the glaze, then pour the remainder into a heatproof container. Cool the butter slightly, then refrigerate until solid, about 2 hours.
To make the cookies: Stir together the flours, salt, and baking soda. Set aside.
Beat the chilled browned butter until soft and creamy. Add the sugars, then the egg, Chai spice, and flavor, beating until light and fluffy. Stir in the flour mixture; then add the water, stirring to blend completely.
Divide the dough in half, flattening each half slightly to make a disk, and wrap in plastic. Refrigerate for at least 1 hour.
Preheat the oven to 350°F. Lightly grease two baking sheets, or line them with parchment.
Working with one piece of chilled dough at a time, place the dough on a lightly floured surface and roll 1/8" to 1/4" thick.
Dip the cookie stamps into flour, then press them into the dough. Using a 2 3/4" round cutter, cut out stamped cookies and transfer them to the prepared pans.
Bake the cookies for 14 to 16 minutes, until lightly browned.
Remove the cookies from the oven, and let them cool on the baking sheet for several minutes, or until set. Transfer the cookies (on their parchment) to a rack to cool completely.
To make the glaze: Combine the reserved butter with the remaining ingredients, adding enough milk to make a thin glaze, about the consistency of syrup.
Brush the glaze onto the cookies and allow to set.