Set your oven to 350°F (175°C). Grease a 9x5-inch loaf pan with nonstick cooking spray and set it aside.
In a large mixing bowl, whisk together flour, baking powder, baking soda, salt, cinnamon, nutmeg, and cloves.
In a medium mixing bowl, whisk together eggs, vegetable oil, vanilla extract, brown sugar, granulated sugar, and applesauce.
Gradually add the wet mixture into the dry ingredients. Stir until just combined, avoiding overmixing.
Gently mix in shredded carrots and chopped pecans or walnuts until evenly distributed.
Pour the batter into the greased 9x5-inch loaf pan and spread evenly.
Place the loaf pan in the preheated oven and bake for 40 minutes, or until the top is golden brown and a toothpick inserted in the center comes out mostly clean.
Let the cake cool in the pan for 30 minutes, then transfer it to a cooling rack to cool completely.
In a stand mixer, beat the softened cream cheese and butter on medium speed for 2 minutes until smooth. Reduce speed to low, gradually adding powdered sugar and vanilla extract, mixing until combined. Increase speed to medium and beat for 1-2 minutes until light and fluffy.
Once completely cool, spread the cream cheese icing evenly over the top. Optionally, garnish with extra chopped nuts or a light dusting of cinnamon.
Enjoy immediately or store the loaf in an airtight container at room temperature for up to 2 days. If frosted, refrigerate for up to 5 days. For long-term storage, wrap the unfrosted loaf in plastic wrap and freeze for up to 3 months.