Heat the oven to 350 degrees F. Melt 3 tablespoons butter in a small pan over medium heat; add maple syrup and brown sugar and cook, stirring, until sugar dissolves. Bring to a boil and cook for another 2 minutes; remove from heat and set aside. When the mixture has cooled a bit, pour it into a 9-inch baking pan and arrange pear slices in an overlapping circle on top.
With a handheld or standing mixer, beat the remaining 8 tablespoons butter and the granulated sugar until light and fluffy. Add vanilla and eggs, one egg at a time, continuing to mix until smooth. In a separate bowl, combine flour, baking powder, and salt.
Add flour mixture to butter mixture in 3 batches, alternating with milk; do not overmix. Carefully spread the batter over pears, using a spatula to make sure it is evenly distributed. Bake until top of cake is golden brown and edges begin to pull away from sides of pan, about 45 to 50 minutes; a toothpick inserted into center should come out clean. Let cake cool for 5 minutes.
Run a knife around edge of pan; put a plate on top of cake and carefully flip it so plate is on bottom and pan is on top. Serve warm or at room temperature.