Step 1: Preheat the oven to 325°F. Butter and line an 8-inch square pan with parchment. Butter the parchment.
Step 2: Toss the rhubarb, 1/2 cup of the granulated sugar, and 1/4 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.
Step 3: 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, about 4 minutes.
Step 4: 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.
Step 5: Spoon the batter over the rhubarb and spread evenly. The pan will be almost full.
Step 6: Bake until a toothpick inserted in the center comes out clean, about 75 minutes.
Step 7: Cool on a wire rack for 15 minutes, then unmold onto a platter. Immediately remove the parchment paper. Cut the cake into squares and serve warm with ice cream.
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