Place a rack in middle of oven; preheat to 350°F. Lightly coat bottom and sides of a 9"-diameter cake pan with oil. Line bottom of pan with a round of parchment paper and smooth to eliminate any air bubbles.
Grate carrots on the large holes of a box grater. Measure out 2½ cups (lightly packed; 250 g).
Set a fine-mesh sieve over a measuring glass and place next to stove. Heat butter in a medium saucepan over medium until foaming, about 2 minutes. Add pistachios and cook, stirring often, until butter browns and nuts smell toasty, about 5 minutes. (Butter will get very foamy and it will be hard to see what’s going on, so once the foam is golden brown, remove pan from heat every 10 seconds to check how the nuts are browning.) Pour mixture through prepared sieve, scraping in all of the browned bits; set nuts and butter aside.
Crack eggs into a large bowl; add brown sugar, granulated sugar, baking powder, baking soda, cardamom, cinnamon, and salt. Using an electric mixer on high speed, beat 3 minutes (mixture should be thick, pale, and increased in volume). Decrease speed to medium-high and gradually stream in reserved brown butter (including browned bits); beat until combined. Coarsely chop reserved pistachios. Using a rubber spatula, fold in flour until almost completely combined, then add nuts and carrots and continue to fold until everything is incorporated. Scrape batter into prepared pan; smooth surface. Place pan on a clean smooth surface and give it a fast 360-degree spin (this pushes the batter up the sides of the pan and minimizes doming during baking).
Bake cake until deeply browned and a tester inserted into the center comes out clean, 50–55 minutes. Let cake cool in pan. Run a small offset spatula or butter knife around perimeter of cake to loosen. Place a plate upside down over cake and invert cake onto plate (leave flat side up). Remove parchment paper.
