In a cast iron skillet or frying pan, melt the butter, maple syrup, sugar, and cinnamon over medium heat. Add the pears, reduce heat to low, and cook for 20 mins on medium-low heat. Remove from heat and set aside.
Make Cake
Preheat the oven to 350℉. Butter a 9-inch square baking pan and line with parchment paper. Arrange the prepared pears in a circular pattern on top of the parchment.
In a medium bowl, whisk together flour, baking powder, baking soda, and salt.
In a large bowl, and using an an electric mixer on medium speed, beat the butter, maple syrup, and granulated sugar until pale and fluffy, 3 to 5 minutes. Add eggs, one at a time, beating well after each addition. Add in vanilla and mix thoroughly.
Reduce mixer speed to low. Add flour mixture in 3 batches, alternating with 2 batches of yogurt and beat until just combined. Transfer batter to prepared pan; smooth top with an offset spatula. Bake until a cake tester comes out clean, about 40 minutes. Transfer pan to a wire rack to cool 10 minutes. Turn out cake onto rack to cool completely.