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Preheat the oven to 350°F
Liberally grease a 9 or 10 inch cast iron skillet, or other pan of similar size, with half of the butter. (Yes, half! It would be sad if your cake stuck to the pan).
Sprinkle the bottom of the skillet with the brown sugar, then add the rhubarb slices in a single layer.
Whisk together the remaining butter, yogurt, eggs, vanilla, and sugar.
In a separate bowl, combine the flour, baking soda, salt, and cinnamon.
Add the liquid mixture to the flour mixture, and stir gently to combine.
Pour the batter into the skillet, on top of the rhubarb, smoothing and leveling the top.
Bake for 50-60 minutes, or until a toothpick stuck in the center comes out clean. Let cool for about 5 minutes.
Gently run a knife around the edge of the cake to loosen it from the skillet. Put a plate that is larger than the skillet over the top. Using hot pads (I highly recommend a hot handle holder for cast iron skillets), carefully flip the cake over and gently lift the pan off the cake.