1. Preheat your oven to 375 F (190 C). Line a 12-cup muffin pan with paper liners or lightly grease each cup.

2. In a large mixing bowl, whisk together the all-purpose flour, granulated sugar, brown sugar, baking soda, cinnamon, nutmeg, ginger, and salt. Make sure there are no lumps and the spices are evenly distributed.

3. In a separate medium bowl, whisk together the sourdough discard, applesauce, vegetable oil, eggs, and vanilla extract until well combined and smooth.

4. Pour the wet ingredient mixture into the dry ingredients. Stir gently with a spatula until just combined. Be careful not to overmix; a few lumps are perfectly fine and help keep the muffins tender. Overmixing can lead to tough muffins.

5. Gently fold in the grated carrots and chopped walnuts or pecans (if using) until they are evenly distributed throughout the batter.

6. Divide the batter evenly among the 12 prepared muffin cups, filling each about two-thirds full. Bake for 20-25 minutes, or until a wooden skewer inserted into the center of a muffin comes out clean. The tops should be golden brown and spring back lightly when touched.

7. Remove the muffin pan from the oven and let the muffins cool in the pan for 5 minutes before transferring them to a wire rack to cool completely. This prevents them from becoming soggy on the bottom.

8. While the muffins cool, beat the softened cream cheese and butter together in a medium bowl with an electric mixer until smooth and creamy. Gradually add the powdered sugar, beating until fully incorporated and fluffy. Stir in the vanilla extract. If the frosting is too thick, add milk one tablespoon at a time until it reaches your desired consistency.

9. Once the muffins are completely cool, spread or pipe the cream cheese frosting generously over the tops. Serve immediately and enjoy!