Preheat oven to 350 degrees. (I always use the convection setting on my oven because it seems to cook more evenly.) Lightly butter the sides of a 9-inch round cake pan.
Combine the 4TBSP butter, brown sugar, cinnamon and nutmeg in a small saucepan and heat to boiling. Remove from heat and scrape mixture into the prepared pan spreading evenly.
Next, arrange the pears over the butter mixture and press down.
Beat the 6TBSP butter and white sugar with an electric mixer until creamy, about 1 minute. Add the eggs and vanilla and beat until very smooth and fluffy, about 2 minutes.
Sprinkle in the flour, cornmeal, baking powder and salt, then beat for 10 seconds. Pour in the milk and beat just until the batter is evenly moistened, about 1 minute.
Pour the batter over the pears then carefully lift the pan and bang back down on the counter once or twice to release any air bubbles.
Bake approximately 35 minutes (watch closely if using convection setting!), or until a knife comes out clean.
Run a knife along the outer edge of the cake to loosen it from the pan. Place a plate over the cake, then flip it over to invert the cake onto the pan.