Preheat oven to 350 degrees Fahrenheit (175 degrees Celsius) and thoroughly grease and flour a 10-inch bundt pan.
In a medium bowl, whisk together the flour, baking soda, baking powder, salt, cinnamon, and ginger.
In a large mixing bowl, beat together the sugar, vegetable oil, eggs, and 1 teaspoon of vanilla extract until smooth and well combined.
Gradually add the dry ingredient mixture to the wet ingredients, stirring until just combined.
Fold in the grated carrots and chopped walnuts using a spatula.
Pour the batter into the prepared bundt pan and smooth the top.
Bake for 50 to 60 minutes, or until a wooden skewer inserted into the center comes out clean.
Allow the cake to cool in the pan for 15 minutes before inverting onto a wire cooling rack to cool completely.
In a small bowl, whisk together the cream cheese, powdered sugar, and milk until smooth, then drizzle over the cooled cake.