Preheat oven to 325 degrees F. Coat a full-size doughnut pan with cooking spray.
In a medium bowl, combine flour, baking powder, and salt. In a small bowl, combine yogurt, brown sugar, egg, oil, and vanilla; whisk until smooth. Add yogurt mixture to flour mixture; stir until well mixed and smooth. Gently fold in raspberries and lemon zest.
Spoon batter into a large resealable plastic bag. Using scissors, cut off a corner; squeeze batter into prepared pan, filling each indent about two-thirds full. Bake about 18 minutes or until doughnuts are nearly firm when lightly pressed. Cool in pan on a wire rack for 3 minutes; turn out doughnuts onto rack.