Preheat the oven to 350°F (180°C). Grease a loaf pan and set aside.
Melt butter in a saucepan over medium heat. Butter will begin to foam and crackle as it melts. When the crackling subsides, the butter will begin to brown. Swirl the pan as the butter cooks. When the butter browns and begins to smell nutty, remove the pan from the flame and transfer the butter to a small bowl. Taking the butter out of the hot saucepan will stop the butter from overcooking and burning. Set aside to cool.
In a large bowl, whisk together flour, sugar, baking soda, salt, cinnamon, ginger, and nutmeg.
*Prepare the buttermilk: add one tablespoon of fresh lemon juice or white vinegar to a measuring cup, pour milk over until you reach ¼ cup (60 ml), stir and let sit for 5-10 minutes before using.
In a medium bowl, whisk together eggs, vanilla extract, and buttermilk. Stir in the grated carrots. When butter has cooled, whisk in the browned butter.
Add the wet ingredients, all at once to the dry ingredients. Fold together and stir until combined, scraping the bottom and sides of the bowl.
Spoon the batter into the prepared loaf pan. Bake for 45 to 50 minutes, or until a skewer inserted into the center of the loaf comes out clean. Remove from the oven and allow to rest in the pan for 15 minutes, before inverting onto a wire rack to cool completely.
For the frosting :
In a medium bowl stir together cream cheese and butter. Add sugar and salt. Whisk in 2 tablespoons of cream, adding more if necessary. The frosting should be thick and spreadable.
Spread the frosting over the cooled loaf cake and sprinkle generously with chopped pecans.