Preheat oven to 350ºF. Grease and flour a 8x8" square cake pan, and line the bottom with parchment paper; set aside.
In a large bowl, whisk together flour, sugar, baking soda, baking powder, ground ginger, and salt.
In a separate bowl, whisk together melted butter, yogurt, eggs, ginger, and vanilla extract. Scrape this mixture into the dry ingredients and stir until nearly combined, with some floury streaks and patches remaining. Fold in the chopped rhubarb with a gentle hand (by this time the floury patches will be incorporated - it's important not to over-mix).
Transfer the batter to prepared cake pan, and arrange remaining rhubarb on top. Bake for 35-45 minutes, until top feels springy when lightly pressed and a toothpick inserted in the centre comes out clean or with just a few moist crumbs. Cool for 10 minutes, then remove cake from pan to finish cooling on a wire rack. Can be served warm with ice cream or at room temperature.