Apricot-Anise Tarts

Combine apricots and 2 cups water in a large heavy saucepan. Bring to a boil, reduce heat to medium, and simmer, stirring and mashing apricots occasionally, until fruit is very soft and broken down and most of liquid is evaporated, about 15 minutes. Add 1/2 cup sugar; stir until dissolved. Remove from heat and stir in preserves and 1 teaspoon almond extract. Let cool completely. DO AHEAD: Can be made 3 days ahead. Cover and chill.
Follow directions for Master Sweet Dough, adding aniseed as directed.
Punch down dough; divide in half. Place half of dough in the center of a 12"-square piece of parchment paper. Using your fingertips, shape dough into a 9" round. Slide dough on parchment paper onto one end of a large baking sheet. Repeat with remaining dough on another piece of parchment; transfer to other end of baking sheet. Divide apricot mixture between rounds, leaving a 3/4" plain border.
Loosely cover tarts with plastic wrap or a kitchen towel and let rise in a warm, draft-free area until puffed but not doubled in size, 45-50 minutes.
Meanwhile, arrange a rack in middle of oven; preheat to 375°F. Beat egg with 2 teaspoons warm water in a small bowl.
Brush border of dough with egg wash. Bake tarts until crusts are golden, 25-30 minutes. Transfer to a wire rack. DO AHEAD: Tarts can be baked 1 day ahead. Let cool completely. Wrap in foil and store at room temperature. Reheat in a 350°F oven until warm, 10-15 minutes. Alternatively, freeze in a resealable freezer bag for up to 1 month. Thaw before reheating.
Dust hot tarts with powdered sugar. Let cool slightly. Using an electric mixer, beat cream, remaining 1/4 cup sugar, and remaining 1/4 teaspoon almond extract in a medium bowl until soft peaks form. Serve tarts warm or at room temperature with almond whipped cream.