Heat oven to 350ºF. Butter and flour a 9-by-3-inch round cake pan. Line the bottom with parchment paper. Set aside.
Put the carrots in a large mixing bowl and set aside.
Put the flour, baking powder, baking soda, spices, and salt in the bowl of a food processor and process for 5 seconds. Add this mixture to the carrots and toss until they are well coated with the flour.
In the bowl of the food processor, combine the sugar, brown sugar, eggs, and yogurt. With the processor still running, drizzle in the vegetable oil. Pour this mixture into the carrot mixture and stir until just combined.
Pour batter into the prepared cake pan and bake on the middle rack of the oven for 45 minutes. Reduce the heat to 325ºF and bake until the cake reaches 205 to 210ºF in the center, about 20 to 25 minutes more.
Remove the pan from the oven and allow cake to cool 15 minutes in the pan. After 15 minutes, turn the cake out onto a wire rack and allow to cool completely before frosting.