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Preheat the oven to 350°F. Lightly grease a 9" x 13" pan.
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Combine the eggs, sugars, and vanilla in a large mixing bowl until smooth.
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Mix the melted butter with the oil. With the beater running, add the oil mixture in a stream, beating until smooth.
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In a separate bowl, whisk together the flour, leaveners, salt, and spices. Add these dry ingredients to the wet ingredients, stirring to make a smooth batter.
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Stir in the carrots, nuts, coconut, and pineapple.
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Spoon the batter into the pan, spreading it to the edges.
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Bake the cake for 40 to 50 minutes, until the cake is golden brown, and a cake tester inserted into the center comes out clean.
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Remove the cake from the oven, and cool in the pan.
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To make the frosting, combine the butter, cream cheese, vanilla, and salt in a medium-sized bowl, and beat together until light and fluffy.
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Add the sugar gradually, beating well.
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Add the milk a little at a time, until the frosting is a spreadable consistency.
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Spread frosting over the cake. Garnish with minced crystallized ginger.