Preheat oven to 350°F. Melt 1/4 cup butter in a 10-inch cast-iron skillet over medium low. Stir in grated ginger. Remove from heat. Sprinkle evenly with brown sugar.

Cut pineapple lengthwise into 8 (1/4-inch-thick) planks, cutting in toward core, and discard core. Cut larger planks into triangles or rectangles. Arrange triangles and smaller planks on top of brown sugar mixture in skillet to fit snugly together (shingling pieces as needed). Set aside.

Beat granulated sugar and 3/4 cup softened butter with a heavy-duty stand mixer fitted with a paddle attachment on medium speed until light and fluffy, 3 to 4 minutes. Beat in vanilla. Whisk together flour, baking powder, salt, and baking soda in a medium bowl. Whisk together eggs and buttermilk in a small bowl. Add flour mixture to granulated sugar mixture alternately with buttermilk mixture, beginning and ending with flour mixture, beating on medium-low speed until just blended before next addition. Spread over pineapple mixture in skillet.

Bake in preheated oven until a wooden pick inserted in center of cake comes out clean, 50 to 55 minutes, tenting with aluminum foil after 40 minutes to prevent excessive browning if needed. Cool in skillet on a wire rack 10 minutes. Invert cake onto a serving platter. Cool 30 minutes to serve warm, or cool completely, about 1 hour.