Preheat oven to 375 degrees. Brush cake pan with oil, line bottom with parchment; oil and flour parchment and pan. Place walnuts in food processor and pulse until finely ground (be careful not to overprocess); set aside.
In a large bowl, whisk together orange zest, flour, baking powder, salt, and ground walnuts. Make a well in the center of flour mixture and stir in eggs, brown sugar, orange juice, and oil until just moistened; do not overmix.
Pour batter into prepared pan and bake until a toothpick inserted in the center of the cake comes out clean, 40 to 45 minutes. Let cool in pan 10 minutes; transfer to a cooling rack to cool completely.
In a small bowl, stir together yogurt and confectioners' sugar. Before serving, dust cake with confectioners' sugar. Serve with sweet yogurt garnished with orange zest, if desired.