Preheat oven to 350°F. Grease a 9-inch square cake pan.
In a medium bowl, whisk together 1 3/4 cups cake flour, 1 teaspoon baking soda, 1 teaspoon cinnamon, 3/4 teaspoon ginger, 1/4 teaspoon nutmeg, and 1/2 teaspoon salt.
In a large bowl or stand mixer, beat 2 eggs, 3/4 cup granulated sugar, and 3/4 cup packed light brown sugar on medium-high speed until pale and fluffy (about 2 minutes).
Mix in 1/2 cup melted coconut oil, 1 1/4 cups pumpkin puree, 1/3 cup milk, and 1 teaspoon vanilla extract.
Reduce speed to low. Add the flour mixture in three additions, mixing until just combined.
Stir in 1 cup chopped walnuts by hand. Batter should be like pancake batter.
Pour into prepared pan. Bake 30–35 minutes, or until a toothpick inserted in the center comes out clean. Cool on a wire rack for 20 minutes.
Serve: Cut cake into squares and top each piece with a dollop of caramel whipped cream. Best served warm.