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.
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.
To make the shortbread: Preheat the oven to 400°F. Lightly grease two baking sheets, or line them with parchment paper.
Stir together the rye flour, brown sugar, baking powder, and salt, breaking up any brown sugar lumps if necessary.
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.
Drizzle in enough milk to bring the dough together with a few gentle stirs.
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.
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.
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.
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.
Remove the cookies from the oven and transfer them to a rack to cool.
Store cookies, well wrapped, at room temperature for several days; freeze for longer storage.