Line baking sheet with parchment paper; coat parchment paper with cooking spray.
Preheat oven to 350°. Stir together apricots and next 3 ingredients in a large bowl. Let stand 30 minutes, stirring occasionally.
Unroll piecrust on prepared baking sheet. Roll into a 12-inch circle.
Drain apricot mixture, reserving liquid. Toss apricots in flour.
Mound apricots in center of piecrust, leaving a 3-inch border. Fold piecrust border up and over apricots, pleating as you go, leaving an opening about 5 inches wide in center.
Stir together egg and 1 Tbsp. water. Brush piecrust with egg mixture, and sprinkle with 1 Tbsp. sugar.
Bake at 350° for 45 minutes or until filling is bubbly and crust is golden. Carefully transfer tart on parchment paper to a wire rack; cool 20 minutes.
Meanwhile, bring reserved apricot liquid to a boil in a small saucepan over medium heat. Boil 1 to 2 minutes or until slightly thickened. Let cool slightly. Brush or drizzle 1 to 2 Tbsp. hot apricot liquid over exposed fruit in center of tart. Serve immediately with remaining apricot syrup, and, if desired, Sweet Cream Topping.