Preheat to 180°C/350°F. Grease a 9”-diameter cake pan with butter or oil.
Cut a parchment round: Fold the paper into a wedge like this. Place inside the pan and smooth to eliminate any air bubbles. Lightly grease the parchment.
The eggs need to be at room temperature for this recipe. If they aren’t, put them in a small bowl and cover them with warm tap water. Let them sit to take the chill off (room-temperature eggs gain more volume than cold eggs do when whipped; cold eggs could result in a tight or dense cake).
Set a fine-mesh sieve over a measuring glass and place next to stove.
Cake
Place butter (1 cup) in a medium saucepan over medium until foaming, about 2 minutes. Add pistachios (1 cup or 130 g) and cook, stirring often, until butter browns and nuts smell toasty, about 5 minutes. The butter will get very foamy and it'll be hard to see what’s going on. Once the foam is golden brown, remove pan from heat and lift out a few nuts with a spoon every 10 seconds to check how they are browning. Pour mixture through prepared sieve. Set butter side.
Add eggs (3) to a large bowl. Add brown sugar (150g), sugar (2 tbsp), baking powder (1¾ tsp), baking soda (¼ tsp), salt (1 tsp), caradmom (1.5 tsp), and cinnamon (½ tsp). Beat with an electric mixer on high speed for 3 minutes until it becomes thick, pale, and slightly increased in volume. Decrease speed to medium-high and slowly stream in reserved brown butter, including browned bits. Beat until combined.
Fold in flour (1.5 cups or 188g).
Coarsely chop reserved pistachios. Fold nuts and shredded carrots (2.5 cups or 250g) the cake batter. Scrape batter into prepared pan; Give the pan a fast 360-degree spin (this pushes the batter up the sides of the pan and minimizes doming during baking).
Place a the pan onto a rack in middle of oven. Bake cake until deeply browned and a tester inserted into the center comes out clean, 50–55 minutes. When in doubt, it’s better to slightly overbake this cake than to take it out too early. Transfer pan to a wire rack and let cake cool in pan.
Run a small offset spatula or butter knife around perimeter of cake to loosen. Place a large plate upside down over cake and invert cake onto plate (leave flat side up; this will give the top of your cake a smooth, even surface and create that dramatic drippage when you glaze it). Remove parchment paper; discard.
Carrot Caramel Glaze
Place a small plate in the freezer.
Cook carrot juice (½ cup), heavy cream (½ cup), sugar (100 g), and salt (1/4 tsp) in a medium saucepan over high heat. Don’t be tempted to stir; the layer of sugar at the bottom of the pan will insulate the cream from the heat. Cook until it ahs a consistency similar to lava, 8–10 minutes. Mixture will initially bubble and foam vigorously and look like it’s going to boil over, but it won’t! Don’t be tempted to reduce the heat; the bubbling will subside and mixture will thicken. You want it to be thick enough to stick to the cake, but do not let it brown.
Test glaze by putting a small dollop onto the chilled plate: When you run your finger through the glaze, it should leave a defined trail. Once the glaze is thick enough, remove pan from heat and stir in butter (2 tbsp). Let glaze cool until you can comfortably hold your finger in it (if it’s too hot, it will run right off the cake), about 15 minutes.