Preheat oven to 350 degrees F (175 degrees C). Grease a 9-inch square baking dish.
Sift all-purpose flour, cake flour, baking powder, and salt together into a bowl.
Beat brown sugar and butter together with an electric mixer in a large bowl until smooth. Beat egg and vanilla extract into the creamy sugar mixture until smooth. Add the flour mixture and stir until just incorporated. Stir strawberries and 1/2 cup marshmallows into the batter; pour into the prepared dish. Sprinkle 2 tablespoons miniature marshmallows atop the batter.
Bake in preheated oven until set in the center, about 25 minutes.