Cook butter and cardamom in a small saucepan over medium heat, stirring occasionally, until butter is light brown and smells nutty, about 6 minutes. Remove from the heat and let stand for 15 minutes, then place in the freezer until just firm, about 15 minutes.
Meanwhile, whisk flour, salt, and baking powder in a small bowl.
Transfer the butter to the bowl of a stand mixer fitted with the paddle attachment and add the sugar. Beat on medium-high speed until pale, 1-2 minutes. Add the egg and beat until combined, scraping down the sides of the bowl as needed. Add the dry ingredients and pistachios and beat on low speed just until combined.
Turn out the dough onto a clean work surface. Divide in half. Place each half on a 9x13" sheet of wax paper with the long end facing you. Roll the dough into 2 logs about 1 1/2" in diameter. Wrap each in wax paper, making sure to cover the ends completely. Chill until firm, at least 1 hour or up to overnight.
Position racks in upper and lower thirds of the oven and preheat to 350°. Let the dough stand at room temperature until soft enough to slice, about 15 minutes. Slice logs into 1/4" rounds. Arrange the rounds on 2 parchment-lined rimmed baking sheets about 1" apart. Bake the cookies, rotating sheets halfway through, until the bottoms are lightly golden, 12–15 minutes. Transfer sheets to wire racks and let cool.
Combine the chocolate and oil in a small heatproof bowl. Heat in 10-second intervals in the microwave, stirring, until melted and combined, 30–40 seconds total.
Dip the cookies halfway into the melted chocolate and transfer to wax paper-lined baking sheets. Sprinkle the chocolate with pistachios and cherries. Chill until the chocolate is set, about 10 minutes.