Make the cake:

1. Position a rack in the bottom third of the oven and heat the oven to 350°F. Butter a 9×13-inch heavy-duty metal cake pan or two 8-inch round cake pans. Line with parchment paper for easy removal. 
2. In a food processor, using the shredder attachment, shred the carrots.  
3. In a large bowl, whisk together the flour, baking soda, salt, cinnamon, and sugars. Add the grated carrots and toss to coat. Add the beaten eggs and oil, and stir with a spatula until evenly mixed. 
4. Scrape the batter into the prepared 9×13-inch pan or divide the batter among the two 8-inch pans. Bake until a toothpick inserted in the center of the cake comes out clean, 45 – 50 minutes for the 9×13-inch pan, about 30 minutes for the round pans. Let cool on a rack to room temperature.
5. If you baked round pans, halve each one crosswise to make 4 round layers total.
6. To make the frosting: Beat the cold cream cheese and sugar together until light and fluffy. Add the heavy cream, a pinch of salt, and the vanilla. Beat again until light and fluffy. 

Frost the cake:

1. If using a 9×13-inch pan: Scrape the frosting onto the center of the cake. With an offset spatula, if you have one, spread the frosting evenly over the surface. 
2. If using round pans: Set one layer on a plate or cake platter. Top with one quarter (roughly) of the frosting, spreading to cover. Top with another layer of cake and repeat this frosting-spreading-layering process until each layer of cake has a nice layer of frosting.