Preheat oven to 350°F. Grease a 9-inch pie dish or 2-quart baking dish with butter.
Spread the blackberries in the dish. Sprinkle evenly with 1 tablespoon granulated sugar.
Whisk eggs with 1/2 cup sugar until pale and slightly foamy.
Add milk, vanilla, lemon zest, and salt. Whisk until smooth.
Whisk in the flour until no lumps remain. Slowly whisk in the melted butter.
Pour the batter over the berries. Tap the dish gently to release bubbles.
Bake 35–40 minutes, until puffed, golden at the edges, and just set in the center.
Cool 10 minutes. Dust with powdered sugar and serve warm or at room temperature.