Brown butter caramel apple cake

Just looking at this cake sends me. This looks absolutely mouth-wateringly delicious.

Submitted by the San Francisco Chronicle.
Ingredients:
The cake:
2 cups cake flour
1/2 teaspoon baking soda
3/4 teaspoon baking powder
1 teaspoon salt
3/4 teaspoon cinnamon
10 ounces (2 1/2 sticks) brown butter, chilled until solid then brought to room temperature
1 cup sugar
1 cup lightly packed golden brown sugar
3 eggs
1 teaspoon vanilla
1/3 cup sour cream
1/4 cup apple cider or juice
2 cups peeled, cored and thinly sliced apples, cut into 1/2-inch lengths
1/2 cup apple butter (see Note)
The frosting:
1/2 cup packed dark brown sugar
3 tablespoons brown butter
1 teaspoon vanilla
1/2 teaspoon salt
1 (14-ounce) can sweetened condensed milk
-- Chopped pecans, for garnish (optional)
Instructions:
For the cake: Preheat oven to 350 degrees. Grease and flour two 9-inch cake pans.
Sift the flour, baking soda, baking powder, salt and cinnamon together, and set aside.
In a stand mixer fitted with a paddle attachment, cream the brown butter with the two sugars until light and fluffy, about 3 minutes.
Add eggs one at a time, then the vanilla, sour cream and cider or juice, mixing well after each addition. After you add the apple juice, the mixture may look curdled, and that's OK. With the mixer on low, gradually add the dry ingredients until just combined, being careful not to over mix. Gently fold in the apple slices.
Divide batter evenly between the two cake pans, and bake until the top springs back and a skewer inserted into the center comes out clean, about 30 minutes. Remove from oven and place on wire racks to cool. After cakes have cooled slightly, run a knife around the edges of the pans and invert cakes out of the pans and onto the racks to cool completely.
When cakes are cool, place one on a cake plate with the flat side up (you may have to cut off a portion of the top to make it level) and cover the top with the apple butter. Place the second cake on top of the first.
For the frosting: Combine the brown sugar, brown butter, vanilla, salt and condensed milk in a medium size, heavy saucepan over medium heat. Bring to a boil, stirring constantly, and cook for another 2-3 minutes until golden brown, smooth and thickened.
Spread the frosting, working quickly, over the top and around the sides of the layer cake. Sprinkle with pecans, if using, and let the icing cool.
Store covered at room temperature for up to 4 days.