Preheat the oven to 325 degrees F. Butter 3 8-inch or 9-inch round cake pans, line the bottom of each with parchment paper, butter the parchment paper, then dust the pans with flour (knocking out any excess) -- or, just use cooking spray. If you have an assortment of round cake pans, that's fine too - I used 2 8-inch and 1 9-inch.

Sift or stir together the dry ingredients in a large bowl, then set aside.

Cream the butter for about 4 minutes until fluffy, using a mixer with the paddle attachment.

Add the sugar, continue beating another 3-4 minutes until light and fluffy again.

Add egg, beat until just combined. Scrape down the bowl.

Alternate adding the flour and applesauce to the mixing bowl in three parts, beginning and ending with the flour. Scrape the bowl, mix for another 10 seconds on low speed.

Divide the batter among your prepared pans and smooth the tops. If you have a kitchen scale to help even distribution of the batter, each pan should contain about 825g of batter.

Bake for about 40-50 minutes, depending on the size of your cake pan (my 9-inch pan took 40 minutes, my 8-inch pans took 50). Rotate the pans halfway through cooking time and be sure an inserted toothpick comes out clean before you remove them. Allow cakes to cool in the pan on a wire cooling rack for 20 minutes. Turn cake out onto the rack, remove parchment paper, allow to finish cooling.

In a heavy-bottomed saucepan, whisk the sugar and flour together. Add the milk and cream, then cook over medium-low heat*, whisking about once a minute, until the mixture has thickened to the consistency of a thin pudding and just come to a boil. This should take about 10-15 minutes.

Immediately transfer the sugar-flour-cream mix into the bowl of an electric mixer and beat on high speed until cooled. This should take about 3-5 minutes.

Reduce speed to low, add the vanilla and butter pieces, keep mixing until fully incorporated.

Raise the speed to medium-high, continue beating until light and fluffy.

Add ⅓ cup of the caramel sauce, mix until combined. This should be about 10 minutes after you transferred from the stove to the mixing bowl.

Place one layer on your serving platter of choice (if you used pans of different sizes, be sure you use the larger sizes first).

If you want, level the top (I usually don't bother). Place about 1 -- 1 ⅓ cups of frosting on top of the layer and evenly spread it across the top.

Place the next cake layer on the frosted layer, repeat step 2.

Place the final layer on top. Spread a very thin layer of frosting across the top and down the sides (this is much easier if you have an angled spatula and/or a rotating cake stand). Put the cake in the fridge to firm up the "crumb coating" you've just applied, which will help reduce loose cake crumbs when you finish frosting the cake. This works surprisingly well, so if you can fit your cake in the fridge, this is absolutely worth doing.

Finish frosting the top and sides of the cake with the remaining frosting**. Drizzle with the remaining 2 tablespoons of caramel sauce. Refrigerate again for 15 minutes to firm up prior to serving.