Serve this cake either warm or at room temperature. Using walnut oil, which contains omega-3 fatty acids, helps cut down on butter and gives the cake a rich flavor.
Preheat oven to 350 degrees. Melt butter with 1/4 cup sugar in a 12-inch cast-iron or large skillet over medium. Arrange pears in a circular fashion, around the edge of the pan. Cook, without turning pears, until pears are golden underneath, 18 to 20 minutes (rotating pan as necessary for even browning).
In a large bowl, combine flours, baking powder, cinnamon, baking soda, and salt; whisk to combine. In a medium bowl, combine remaining sugar, eggs, buttermilk, applesauce, oil, and vanilla; whisk to combine. Add wet mixture to dry mixture and whisk until just combined.
Pour batter over pears; spread to cover. Transfer to the oven; bake until a toothpick inserted into cake comes out clean, 25 to 30 minutes. Allow to stand for 5 minutes. Run a knife around the edge and turn out onto a serving platter.