Preheat the oven to 375°. Generously butter an oval or square baking dish, about 8" × 8"× 2".
For the crust: Combine flour, sugar, baking powder, and salt in a large mixing bowl. Stir to mix the dry ingredients thoroughly. Make a well in the center of the dry ingredients and add the beaten egg. Stir mixture with a fork until it gets crumbly, the consistency of cornmeal. If mixture is too dry, add a little more of another beaten egg.
For the filling: Place the berries into another large mixing bowl. Stir the sugar and flour together, then sprinkle over the fruit and carefully toss until it coats the fruit thoroughly.
Pour the berries into the buttered baking dish and spread evenly, and pat them gently with your hands.
Sprinkle the crumbly flour mixture evenly over the top of the fruit. Drizzle the melted butter evenly over it. Place in the oven and bake about 40 minutes, or until the topping is golden.
Serve warm with whipped cream or ice cream. Or serve at room temperature, and the crust will have absorbed even more of the berries' juices.