Preheat the oven to 325°F. Butter and line an 8-inch square pan with parchment. Butter the parchment.
Toss the rhubarb, ½ cup of the granulated sugar, and ¼ cup of the brown sugar together in a bowl. Layer the rhubarb across the bottom of the pan in a grid pattern, adding any remaining sugar mixture to the pan.
Whisk together the flour, baking powder, and salt. In a separate bowl, beat the butter, remaining sugars, and lemon zest with an electric mixer until light and fluffy, 4 minutes. Scrape down the sides of the bowl. Add the vanilla and eggs, one at a time, beating after each addition.
Stir in the yogurt on low speed. (The mixture may look curdled.) Gradually add the flour mixture and stir to combine.
Spoon the batter over the rhubarb and spread evenly. (The pan will be almost full.)
Bake until a toothpick inserted in the center comes out clean, 1 hour, 15 minutes. Cool on a wire rack for 15 minutes, then unmold onto a platter. (Discard the parchment paper.) Cut the cake into squares and serve warm with ice cream.
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