Preheat your oven to 325°F. Spray three 8-inch cake pans with nonstick spray, line with parchment paper, and spray again. Set aside.

In a large mixing bowl, Whisk the packed dark brown sugar and vegetable oil. Whisk by hand well until the sugar is fully incorporated into the oil. About 2 minutes.

Add the eggs to the sugar-oil mixture, one at a time, mixing well after each addition.

Stir in the smooth unsweetened applesauce and pure vanilla extract until well combined.

In a separate bowl, sift together the all-purpose flour, baking powder, baking soda, salt, ground cinnamon, and ground ginger. (This removes the clumps from the flour)

Gradually add the dry ingredients to the wet ingredients, mxing until just combined. Be careful not to overmix.

Switch to a rubber spatula, gently fold in the finely grated carrots until evenly distributed throughout the batter.

Pour the batter into the prepared cake pans and spread it evenly with a spatula.

Bake in the preheated oven for 35-38 minutes, or until a toothpick inserted into the center comes out clean. While the cake is baking, start on making the caramel and brown butter frosting.

Once baked, remove the cake from the oven and allow it to completely cool in the pan.

In a medium heavy-bottomed saucepan, combine the sugar, water, and corn syrup. With a small spoon or rubber spatula, gently stir to moisten all the sugar, then stop stirring.

Set the pan over medium heat and cook undisturbed for 5–10 minutes, until the mixture turns deep amber. This does happen fast so keep an eye. The mixture will bubble really fast at first, as it starts to caramelize the bubbles will get bigger and slow down.

Meanwhile, warm the heavy cream in the microwave for about 1 minute.

Once the caramel is a rich copper color, remove from heat and carefully pour in the warm cream.
Caution: The mixture will bubble up rapidly. Pour slowly and whisk constantly.

Whisk until smooth. Stir in the vanilla, salt, and butter. Return to low heat briefly if needed to smooth it out.

Let cool slightly, then transfer to a heat-safe jar and refrigerate until ready to assemble. The caramel will thicken as it cools.

In a medium saucepan, melt 1½ cups (3 sticks) of unsalted butter over medium heat. Continue cooking, swirling the pan occasionally, until the butter foams and turns a deep amber color with a nutty aroma — about 5–7 minutes.

Let the butter cool to room temperature (about 2 hours) before using.

When ready to frost, place the room temperature browned butter and cream cheese into the bowl of a stand mixer fitted with the paddle attachment. Beat on medium-high speed for 2–4 minutes until light and fluffy.

Add the powdered sugar, heavy cream, vanilla, and salt. Mix on low to combine, then beat on high for 2 more minutes.

Use a rubber spatula or wooden spoon to beat out any air pockets before frosting.

Make sure the cake, caramel and frosting have completely cooled before you assemble. If you assemble while any of it is warm the cake might slip off.

Place a small dollop of frosting in the center of a cake board to act as glue.

Add the first cake layer top-side up. Spread about 1 cup of frosting over the top and drizzle with caramel sauce.

Add the second cake layer top-side down and repeat the frosting and caramel.

Add the final cake layer top-side down again.

Apply a thin crumb coat of frosting and chill the cake for 15 minutes.

Frost the entire cake with the remaining brown butter frosting. Add a caramel drip if desired and serve extra caramel on the side. If you want to add walnuts, now is your time! 
This cake is best served at room temperature for the perfect texture and flavor. I highly recommend it.