Preheat the oven to 340°F/170°C and line a 9x5-inch/23x13 cm loaf pan with parchment.

Whisk eggs, sugar, olive oil, vanilla, ginger, and orange zest until smooth. Add grated carrots and chopped nuts.

In a separate bowl, whisk flour, baking powder, cinnamon, and salt, then fold into wet ingredients.

Pour batter into the pan and bake for 35–40 minutes, until a skewer comes out with a few moist crumbs. Cool completely.

Make brown butter at least 1 hour before frosting: melt unsalted butter over medium heat, stirring, until foaming. Cook until milk solids turn amber and smell nutty, 5–7 minutes. Chill until firm.

Beat cream cheese, powdered sugar, and chilled brown butter until smooth and fluffy.

Slice cake horizontally, spread frosting between layers, and cover top with remaining frosting. Garnish with citrus zest. Store wrapped in the refrigerator.