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To make the brown butter: Melt, then cook the butter in a saucepan over medium heat, swirling occasionally, until golden brown, 5 to 10 minutes.
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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.
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To make the cookies: Stir together the flours, salt, and baking soda. Set aside.
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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.
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Divide the dough in half, flattening each half slightly to make a disk, and wrap in plastic. Refrigerate for at least 1 hour.
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Preheat the oven to 350°F. Lightly grease two baking sheets, or line them with parchment.
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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.
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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.
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Bake the cookies for 14 to 16 minutes, until lightly browned.
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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.
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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.
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Brush the glaze onto the cookies and allow to set.