Step 1: Preheat your oven to 350°F (175°C) and grease a 9-inch square or round baking pan with oil or butter.
Step 2: In a large mixing bowl, whisk together the flour, 1.5 cups sugar, baking soda, baking powder, salt, and cinnamon until well combined.
Step 3: In a separate bowl, beat the eggs and vegetable oil together until smooth, then stir in the finely grated carrots.
Step 4: Gradually fold the wet ingredients into the dry ingredients, being careful not to overmix. Gently stir in the toasted walnuts.
Step 5: Pour the batter into the prepared pan and bake for 40-45 minutes, or until a skewer inserted into the center comes out clean.
Step 6: While the cake is baking, prepare the syrup by combining water, 0.5 cup sugar, and fresh ginger slices in a small saucepan over medium heat.
Step 7: Bring the syrup to a boil, then reduce heat and simmer for about 15 minutes until it thickens slightly and becomes aromatic. Strain out the ginger pieces.
Step 8: Once the cake is removed from the oven, poke small holes across the surface with a toothpick and slowly pour the warm ginger syrup over the top. Serve your warm carrot cake with ginger syrup immediately while still warm.