Easter Bread Dolls (Pupi or Titola)

Line two large baking sheets with parchment paper. Butter paper.
Cut off 1/3 of ball. From smaller piece, pinch off golf-ball-sized piece. On lightly floured surface, roll largest piece of dough into 20-inch-long rope. Roll medium-sized piece into 10-inch-long rope. Divide smallest piece into 2 pieces and roll each into thin 5-inch-long rope.
On floured surface, form longest rope into inverted "U" shape. Arrange medium-sized rope with 1 end intersecting top center of "U," to form 3 strands of equal length. Gently press where ropes intersect to adhere.
Gently braid strands, tucking last inch under to form round bottom. Gently press 1 dyed egg into braid about 1 inch from top. Using remaining 2 thin ropes, form cross over egg, pressing ends to adhere to body of doll. Using tines of fork, make 4 crimps on bottom of doll to form "feet." Using spatula or bench scraper, carefully transfer doll to baking sheet, leaving room for second doll alongside.
Repeat entire process with remaining 2 balls of dough to make 2 more dolls, arranging third doll by itself on 2nd baking sheet. Loosely cover dolls with buttered plastic wrap. Let rise in warm, draft-free place until doubled in bulk, about 45 minutes to 1 hour.
Preheat oven to 325°F. Bake dolls 35 minutes. In small bowl, whisk together egg, sugar, and 2 tablespoons water until sugar is dissolved. Brush dolls with egg mixture and bake until very deep golden brown and knife inserted into center comes out clean, about 10 minutes more. Transfer to wire rack and cool completely. (Dolls will keep, wrapped well in plastic wrap, 1 week at room temperature or 6 to 8 weeks frozen.)