Whirl 2 1/4 cups flour, the cornmeal, 5 tbsp. sugar, the baking powder, and salt in a food processor to blend. Add butter to bowl and freeze bowl 20 minutes. Pulse until butter is a little smaller than pea-size. Add 1/4 cup ice water and pulse until evenly distributed (about 10 times); mixture will look dry.
Pour ingredients onto a work surface. Press and squeeze with hands until dough holds together. Flatten into a 6-in. disk, wrap in plastic, and chill in refrigerator 1 hour. Set out twelve 4-in. flan molds or tart pans, preferably nonstick (butter and flour pans if they aren't nonstick).
Preheat oven to 375°. On a lightly floured work surface, roll dough a bit bigger than a 12- by 16-in. rectangle, lifting it and reflouring surface as needed to prevent sticking. Cut 12 even 4-in. squares. Gently set each square into a flan or tart pan. Set pans on 2 rimmed baking sheets. If needed, chill until dough is cool but still flexible, 5 minutes.
In a medium bowl, gently combine berries and 1/2 cup sugar. Let stand until sugar looks wet but before syrup starts to pool, stirring occasionally, about 5 minutes.
Scoop 1/3 cup strawberries onto each dough square. Fold dough corners over berries by 1/2 to 1 in. Brush corners with milk and sprinkle them lightly with sugar.
Bake crostatas until pastry is deep golden brown and crisp, about 35 minutes, switching pans halfway through. Set baking sheets on a rack and let crostatas stand until just cool enough to touch, 10 minutes. Loosen crusts from pans with a knife. Carefully invert each pastry onto your hand, then set right side up on a plate. Serve warm or cool, with whipped crème fraîche.
*Find at jbprince.com.