Preheat the oven to 350 degrees F.
Using an electric mixer at medium speed, combine the eggs, sugar, oil and pumpkin until light and fluffy. Stir together the flour, baking powder, cinnamon, salt and baking soda. Add the dry ingredients to the pumpkin mixture and mix at low speed until thoroughly combined and the batter is smooth. Spread the batter into a greased 13 by 10-inch baking pan. Bake for 30 minutes. Let cool completely before frosting. Cut into bars.
To make the icing: Combine the cream cheese and butter in a medium bowl with an electric mixer until smooth. Add the sugar and mix at low speed until combined. Stir in the vanilla and mix again. Spread on cooled pumpkin bars.
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