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Heat the oven to 350 degrees. In a medium bowl, whisk together the flour, baking powder and salt. Generously brush a 9-inch round cake pan with oil.
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In a blender or food processor, finely chop the carrots by pulsing about 6 to 10 times until minced. Add the sugar and pulse until just combined, about 30 seconds. Add the oil and eggs all at once. Pulse until the mixture is combined and looks a bit foamy, 30 seconds. (You can also do this step without a blender or food processor: Grate the carrots into a medium bowl using the fine side of a box grater. Add the sugar, eggs and oil, and use a whisk to combine.)
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Pour the wet mixture into the dry ingredients, and, using a whisk, mix until just combined. Gently mix in the sour cream and pour the mixture into the prepared cake pan. Bake until the sides pull away slightly from the edge of the tin and a toothpick inserted into the center comes out clean, about 45 to 50 minutes.
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Move the cake to a baking rack set in a sheet pan and let it cool slightly in the pan, about 10 minutes. Invert the cake onto the baking rack and let cool completely.