Heat oven to 350°. Line an 8-inch square baking pan with foil, letting ends extend above pan on 2 sides.

Shortbread: Process flour, sugar and salt in a food processor to mix. Scatter butter on top and process just until a dough forms. Press dough gently over bottom of prepared pan. Pour egg white over dough and tilt pan until egg white covers dough completely, then pour off excess. Bake until golden brown, 20-25 minutes.

Meanwhile prepare filling and topping: Bring 3/4 cup of the cherry juice, the sugar and salt to a boil in a medium saucepan. Whisk cornstarch into remaining 1/4 cup (cold) cherry juice, then whisk into boiling mixture until mixture thickens, boils and turns clear, 3-4 minutes. Remove pan from heat and stir in cherries.

Topping: Process flour, both sugars and salt a few seconds to mix. Add oats and butter and process a few seconds until evenly blended. Remove base from oven and increase oven temperature to 425°. Spread filling evenly over base, then sprinkle topping evenly over filling.

Bake until topping is lightly browned, about 25 minutes, then broil until topping is a slightly deeper brown, watching carefully that it doesn't scorch. Cool completely in pan on a wire rack before lifting foil by ends onto a cutting board. Cut in 16 squares; remove from foil.