Preheat oven to 350°F. Spray three 8-inch round or four 6-inch round cake pans with non-stick cooking spray and line the bottoms with parchment paper rounds. Spray again and set aside.
In a medium bowl, combine the flour, spices, baking powder, baking soda and salt. Whisk to combine and set aside.
In the bowl of a stand mixer fitted with the paddle attachment, or in a large bowl with a hand mixer, beat the oil and sugars on medium speed until combined (about 2 minutes).
Add the pureed peaches and vanilla. Stir to combine.
With the mixer on medium speed, gradually add the eggs one at a time. Make sure to scrape down the bowl between each egg addition. Once all of the eggs are combined, beat on medium speed for about 1 minute to fully combine.
With the mixer on low speed, gradually add the flour mixture.
Fold in the grated carrots.
Evenly divide the batter between cake pans. Bake for 27-30 minutes, or until a toothpick inserted in the center comes out clean.
Let cakes cool in pans on a wire rack for 10-15 minutes. Invert the cake layers onto the cooling racks to cool completely.
Level cake layers if needed using a serrated knife.