
Begin browning the butter. When making browned butter, it is best to use a stainless steel pan so that you can see the color of the butter change. Heat the butter in a large shallow frying pan over medium heat until melted. Swirl the pan around a bit every couple minutes to help it cook evenly. Over a period of several minutes, you'll notice the foam at the top of the butter start to change from light yellow to a dark tan. Once it reaches the dark tan stage and the butter looks light brown and golden, smell it. It should smell nutty and similar to toffee.
Remove it from the heat and allow it to cool slightly before pouring it into a small bowl lined with plastic wrap. Place it in the refrigerator to harden. Now, you have two options. You could keep poking at the butter while it's in the fridge until it feels firm but still soft enough to be made into a buttercream, or, rather than continually poking at the butter, just refrigerate it until it's hard, then take it out and leave it at room temperature until it's softened a bit.
While the brown butter is in the fridge, you can start making the cake. Preheat the oven to 350 degrees Fahrenheit. Mix together the flour, corn meal, baking powder, baking soda, and salt in a large bowl until well blended. Set aside. In an electric mixer fit with a paddle attachment, beat together the eggs and sugar at medium speed until thick, about 1 minute. Add the olive oil, cider, and vanilla extract and mix until combined. Add the flour mixture to the batter in thirds until completely incorporated, then stir in the rosemary. Evenly distribute the batter between 3 well-greased and lightly-floured 8-inch cake pans. Place the pans in the oven and bake for 30-35 minutes, or until the tops of the cakes turn golden brown and a toothpick inserted into the cake comes out clean. Remove the cakes from the pans and place on a wire rack to cool.
While the cakes are cooling, you can finish preparing the buttercream. Beat together the softened brown butter, powdered sugar, and honey until a smooth buttercream forms. It is okay if there's little burnt brown butter bits in it, because these little guys taste amazing and add a pretty speckled coloring to the frosting. Once blended, set aside. When the cakes are finished cooling, you can layer and frost them with the buttercream and serve.