Start by setting your oven to 350°F so it's fully heated by the time your batter is ready. While it warms up, lightly grease your 9-inch round cake pan even though the caramel will coat most of it. This just gives you a little insurance against sticking around the edges.
In a saucepan over low heat, melt the butter and brown sugar together, stirring often until it turns smooth and glossy. I've microwaved this in 30-second bursts before, but low heat on the stove gives me more control. Pour this warm caramel straight into your prepared cake pan.
Lay your thin pear slices over the caramel, working from the center outward in a spiral pattern. I like to overlap them slightly so there are no caramel gaps once it bakes. This step is where the cake gets its signature pretty top, so take an extra minute here.
In a separate bowl, whisk together the flour, sugar, baking powder, and salt. Getting these evenly distributed now means you won't end up with pockets of baking powder in your finished cake. Set this bowl aside while you work on the wet mixture.
In a large bowl, whisk the milk, oil, egg, and vanilla until just blended. I don't overthink this step a fork works fine if you don't feel like pulling out a whisk. You just want everything loosely combined before the dry ingredients go in.
Add the dry mixture into the wet mixture and stir until just incorporated. I always stop as soon as I don't see streaks of flour anymore. Overmixing here is the fastest way to end up with a dense, rubbery cake instead of a soft one.
Pour the batter gently over the pears and caramel, spreading it evenly to the edges. Place the cake on the middle rack with a cookie sheet on the rack below to catch drips. Bake for 50 to 60 minutes, until the top is golden and a toothpick comes out with a few moist crumbs.
Let the cake rest on a wire rack for about 10 minutes before flipping it. Any sooner and the topping can slide apart; any later and the caramel may cool and stick to the pan. Place your serving plate over the pan, flip confidently, and lift the pan away.