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Preheat the oven to 350°F. Grease and flour a 9" x 13" pan or two 8" x 2" round pans.
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To make the cake: Combine the oil and sugar, and stir in the starter. Add the eggs one at a time, beating well after each.
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Fold in the pineapple, carrots, walnuts, coconut, and vanilla.
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In a separate bowl, combine the flour, cinnamon, salt, and baking soda.
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Add the dry ingredients to the wet ingredients, stirring just to combine. Pour the batter into the prepared pan.
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Bake the cake for 45 minutes, or until a toothpick inserted into the center comes out clean, and the sides are just beginning to pull away from the edge of the pan.
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Remove the cake from the oven and cool it completely on a rack before frosting.
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To make the frosting: Combine the butter, cream cheese, and vanilla; beat until light and fluffy.
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Add the sugar gradually, beating to combine.
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Stir in the nuts and/or ginger. Add the milk, a little at a time, until the frosting is a spreadable consistency.
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Serve the cake at room temperature. Store it in the refrigerator for up to 5 days. Unfrosted, the cake can be frozen, well wrapped, for up to 3 months. Frost just before serving.