Mix together apricots, white sugar, and rum in a bowl. Cover and let macerate in the refrigerator for 8 hours, or overnight.

Preheat the oven to 400 degrees F (200 degrees C). Line a baking sheet with a 12-inch square of parchment paper. Spray the parchment paper with cooking spray.

Place flour, cold butter, dark brown sugar, baking powder, salt, nutmeg, and ginger into a food processor. Blend in short bursts until butter is in pea-sized lumps.

Whisk together buttermilk and egg in a bowl; pour into butter-flour mixture and continue to blend until it forms a soft dough. Form dough into a smooth ball, using additional flour as needed.

Pat dough ball into a thick circle onto the prepared baking sheet.

Drain apricots, reserving juice in a bowl. Spread apricots evenly on top of dough circle; draw edges of dough up over fruit.

Bake in the preheated oven until crostata is baked through, golden on top, and fruit is soft, about 45 minutes.

While crostata is baking, pour reserved apricot juice and orange juice into a saucepan. Bring mixture to a simmer; cook syrup until it has reduced by 2/3.

Remove crostata from the oven. Brush syrup on top. Let cool slightly. Sprinkle with powdered sugar and serve.