Brilliant Banana Upside-Down Cake

For topping
¾ stick unsalted butter
1 cup packed light brown sugar
3 large just-ripe bananas, cut into 1 1/2-inch pieces

For batter
1 ¾ cups all-purpose flour
2 teaspoons baking powder
¼ teaspoon cinnamon
¼ teaspoon salt
½ stick unsalted butter, softened
1 cup granulated sugar
2 large eggs
1 teaspoon pure vanilla extract
¾ cup warm milk

Special equipment: a well-seasoned 10-inch cast-iron skillet

Directions
Preheat oven to 350°F with rack in middle.
Melt butter in skillet over medium heat, then stir in brown sugar and simmer, stirring, 4 minutes. Remove from heat and arrange bananas on top of sugar mixture in concentric circles, fitting in as many pieces as possible.
Whisk together flour, baking powder, cinnamon, and salt. Beat together butter and sugar with an electric mixer at high speed until pale and fluffy, about 3 minutes. Add eggs, 1 at a time, beating well after each addition. In a separate bowl, mix warm milk with vanilla. Reduce speed to low and add flour mixture in 2 batches, alternating with milk, mixing until just incorporated.
Spoon batter over bananas, smoothing top. Bake until top is golden and a wooden pick inserted into center comes out clean, 40 to 45 minutes.
Let cake stand in skillet 15 minutes. Put a plate over skillet and invert cake onto plate (keeping plate and skillet firmly pressed together). Replace any banana pieces stuck to bottom of skillet.
Serve cake slightly warm or at room temperature.