In a seasoned 10 inch cast iron skillet, melt butter and medium heat. Spread the butter out to cover the skillet. Add the brown sugar and maple syrup. Simmer, whisking until sugar is dissolved, about 2 minutes. Remove from heat. Arrange pear slices decoratively over the sugar and butter mixture. Sprinkle walnuts over and around pears. Set aside and mix cake batter.
For the cake:
In a large bowl, whisk flour, cinnamon, baking soda, and salt. Set aside. In another bowl, combine greek yogurt, vanilla, and walnut extracts. Set aside. In the bowl of an electric mixer, beat butter until smooth. Add sugar and beat until light and fluffy, about 5 minutes. Beat in the 1/3 cup maple syrup and mix 2 to 3 minutes. Turn mixer to low, add eggs one at a time. Beat well after each addition. Reduce the speed and add dry ingredients alternating with greek yogurt beginning and ending with dry ingredients. Scrape bowl if necessary. Beat only long enough to incorporate the ingredients, over-beating will result in a tough cake.
Preheat oven to 350 degrees fahrenheit.
Carefully drop batter by spoonfuls over pears in skillet. Spread evenly, careful not to move the pears. Bake at 350 degrees 40 to 45 minutes. Test with a wood pick in center. Wooden pick should come out clean when cake if done. Cool the cake in the skillet on a rack for 10 minutes.
Run a knife around the edge of cake. Carefully invert cake onto a platter. I've found wearing oven mittens is the easiest way to do this without slipping. Remove the skillet and allow cake to cool on platter on wire rack.
Serve warm or at room temperature with vanilla ice cream.