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To brown the butter: Melt the butter in a saucepan over medium heat. Once melted, the butter will start to sizzle and may spatter. Continue to cook the butter, swirling the pan regularly, until it’s a rich golden brown and brown bits start collecting at the bottom of the pan; the butter will have stopped sizzling and may also have a layer of foam on the surface.
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Remove the pan from the heat and pour the butter into a heatproof bowl. Cool the butter slightly at room temperature, then transfer it to the refrigerator to chill until it solidifies but isn’t rock-hard, about 30 minutes.
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To make the shortbread: Preheat the oven to 400°F. Lightly grease two baking sheets, or line them with parchment paper.
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Stir together the rye flour, brown sugar, baking powder, and salt, breaking up any brown sugar lumps if necessary.
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Remove the brown butter from the refrigerator, stir it briefly to bring it to an even temperature, then work it into the dry ingredients until everything is crumbly.
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Drizzle in enough milk to bring the dough together with a few gentle stirs.
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Gather the dough into a ball. It should be entirely cohesive; if dry bits are falling off around the edges and everything isn’t holding together nicely, add a bit more milk.
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Place it on a lightly floured work surface, and flatten it to a rough square. You can roll the dough immediately, without chilling; or shape it, wrap it in plastic, and refrigerate it for up to a couple of days. Take the dough out of the refrigerator 15 to 20 minutes before rolling, to warm it up.
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Roll the dough into a 1/8”-thick round. Use your favorite cookie cutters or stamps to cut out cookies that are roughly 2” to 4” in diameter. Re-roll scraps to cut out more cookies. For simplest cutting (and with no dough scraps to re-roll), roll the dough into a square or rectangle and cut square cookies using a square cutter, bench knife, table knife, or pizza wheel.
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Transfer the cookies to the prepared baking sheet and bake for 7 to 12 minutes (larger cookies will take longer), or until golden brown around the edges and on the bottom; you’ll want to watch them toward the end of the baking time as they can go from perfectly golden to too dark within a couple of minutes.
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Remove the cookies from the oven and transfer them to a rack to cool.
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Store cookies, well wrapped, at room temperature for several days; freeze for longer storage.
