Preheat the oven to 350 degrees F.
Generously butter the sides of a 10-inch cast iron skillet. Combine the softened butter and brown sugar and spread them evenly on the bottom of the skillet.
Slice the plums into thin wedges and set aside. Do not add to the skillet yet. See notes below.
In a large mixing bowl, whisk the flour, ginger, baking powder, baking soda, and salt.
Use a microplane grater to zest the lemon. In another large bowl, whisk together the melted butter, sugar, vanilla and lemon zest.
Whisk the eggs into the melted butter mixture one at a time.
Finally, add the yogurt to the liquid ingredients and whisk until completely incorporated.
Arrange the plum slices in the skillet on top of the brown sugar mixture. It's important to do this before finishing the batter. See notes below.
Add the dry ingredients to the wet in two batches, using a silicone spatula to combine the ingredients. Stir gently, smoothing the batter until there are no lumps of flour remaining. The batter will be thick, but full of airy bubbles from the activation of the rising agents.
Pour the batter over the plums in the skillet and gently spread evenly. Bake for about 40-45 minutes until the cake is deeply golden brown and is pulling away from the sides of the skillet, and a tester inserted into the center of the cake comes out clean. Remove from oven and immediately run a knife or offset spatula around the outside edge of the cake.
Let the cake cool in the pan for about 10 minutes, then cover the skillet with a large serving plate and carefully flip the cake out onto the plate. Let cool for 30 to 60 minutes before serving.