Make the topping: Melt butter in the bottom of a 10 1/2-inch cast-iron skillet over low heat. Add brown sugar, stirring until dissolved. Swirl to coat the bottom; remove from heat, and cool. Cut pears into 1/4-inch-thick wedges, and arrange them in a circular pattern over the brown-sugar mixture to cover completely; set aside.
Make the batter: Preheat the oven to 350 degrees. In a medium bowl, whisk together flour, baking powder, and salt; set aside.
In the bowl of an electric mixer fitted with the paddle attachment, cream butter and sugar on medium speed until light and fluffy, about 2 minutes. Add vanilla extract; beat to combine. Add egg yolks, one at a time, beating to combine.
Alternating with the milk, gradually add the flour mixture to the butter mixture, and mix, on low speed, just until the flour is incorporated.
In a large bowl, beat reserved egg whites and the cream of tartar with a hand mixer until stiff but not dry. Using a rubber spatula, fold egg whites into the batter. Transfer to skillet. Using an offset spatula, spread the batter evenly, being careful not to disturb the pears.
Bake until well browned on top and a cake tester inserted into the center comes out clean, about 45 minutes. Run a knife around inside of pan, and immediately invert the cake onto a serving dish. Serve warm or at room temperature.
When spreading the batter, be careful not to disturb the pears, which, after the cake is baked and inverted and the fruit is caramelized, will grace its top much in the way apples do a tarte Tatin.