Preheat oven to 350. Generously butter or spray a 9-inch springform pan or other similar-sized pan (See notes!)
In a large mixing bowl, combine the flour, baking powder, baking soda, salt, spices and brown sugar. Stir to combine.
Make a well in the middle of the bowl and add the eggs, olive oil, vanilla, applesauce, honey, and milk. Mix together gently and then incorporate into dry ingredients, mixing until just combined, with no streaks or dry areas.
Fold in diced apples .
Spoon into prepared pan and smooth with the back of a spoon. Bake on a middle shelf for 33-38 minutes, or until a toothpick inserted into the middle of the cake comes out with a few moist crumbs.
Allow cake to cool on a wire rack before frosting. (You can release from the springform after 10-15 minutes to cool it more quickly. Just run a knife around the edge first.)
Pour the browned butter into a bowl and whisk in the 2 cups sifted powdered sugar and vanilla, stirring until smooth. Add the milk, a tablespoon at a time until the desired consistency is achieved. (If you want a thick frosting, add less milk, for a pourable glaze, add milk until it's smooth and drizzles nicely off the spoon.)
Spread icing over the cake and serve warm, at room temperature or even cold from the fridge. (This cake tastes even better on days 2 and 3!)
A note on the baking pan: No springform pan? No problem. You can use a 9x13 baking pan, a 10 inch round baking pan, or even a bundt pan. Just adjust baking time if needed, the thinner the batter, the less baking time will be.
Base recipe adapted slightly fromĀ Simple Cakes by Odette Williams