In this update of a retro dessert, a judicious amount of caramel coats the bottom of the pan before the fruit and batter are added. If you have a melon baller, pull it out—it's the perfect tool for coring pears.
Preheat oven to 350°F.
Heat 2 tablespoons sugar in a medium cast-iron skillet over medium heat, stirring occasionally with a wooden spoon, until it melts and turns a deep amber color, 5 to 10 minutes. Remove from heat. Immediately add 2 tablespoons butter, stirring vigorously. When the butter is incorporated, spread the caramel in an even layer. Shingle pear wedges on top.
Whisk cake flour, whole-wheat flour, baking powder, baking soda and salt in a medium bowl. Combine oil with the remaining 1/2 cup sugar and 4 tablespoons butter in a large bowl. Scrape in vanilla bean seeds. Beat with an electric mixer on medium speed until smooth. Add eggs one at a time, beating well after each. Add yogurt and beat until blended. Gradually add the flour mixture, beating on low speed just until blended and scraping the sides as needed. Spoon the batter over the pears and spread with an offset spatula to cover.
Bake the cake until golden brown and a toothpick inserted in the center comes out clean, 25 to 30 minutes. Let cool in the pan on a wire rack for 10 minutes.
Run a knife around the edge to loosen. Carefully pour out any excess liquid from the pan into a measuring cup. (It's OK if there isn't any excess liquid—it all depends on how juicy your fruit is.) Carefully invert the cake onto a serving plate and drizzle with any reserved liquid. Let cool slightly, about 10 minutes.
Store, well wrapped, at room temperature for up to 2 days.