Combine water and sugar in bowl of heavy-duty mixer. Stir in yeast. Let stand until yeast dissolves, about 10 minutes. Using rubber spatula, mix in flour (dough will be firm). Cover bowl with plastic wrap. Let starter rise until puffy, about 45 minutes. (Initially the starter is firm and compact, but it softens and becomes puffy and spongy after rising.)
For step 2:.
Attach dough hook to mixer. Add all ingredients in step 2 to starter. Beat until blended, scraping down sides of bowl often, about 5 minutes (dough will be soft and thick). Scrape dough off hook; remove hook. Cover bowl with plastic. Let dough rise at room temperature until puffy and bubbly on top, about 1 hour. The dough will look thick, shiny, and slightly puffed.
For step 3:.
Reattach clean dough hook. Add first 5 ingredients in step 3 to dough; beat until blended. Add flour. Beat at low speed until smooth, scraping down bowl and hook often, about 5 minutes (dough will be firm and compact). Scrape dough off hook; remove hook. Cover bowl with plastic; let dough rise at room temperature until lighter in texture and slightly puffed, about 3 1/2 hours. The dough will double in volume and become lighter in texture but less glossy.
For step 4:.
Reattach clean dough hook. Mix water and yeast in small cup. Let stand until yeast dissolves, about 10 minutes; add to dough. Add 1 1/3 cups flour, half of butter, sugar, and 2 yolks; beat until dough is smooth, about 3 minutes. Scrape down dough hook and sides of bowl. Add remaining 2 yolks, milk, vanilla extract, and salt. Beat at low speed until blended, about 3 minutes. Scrape down hook. Add remaining 2/3 cup flour, remaining butter, and orange peel. Beat dough until well blended, about 5 minutes. Scrape dough into very large (at least 4-quart) buttered bowl. Cover with plastic. Let dough rise at room temperature until doubled and indentation remains when 2 fingers are pressed about 1/4 inch into dough, 8 to 10 hours.
For step 5:.
Sprinkle 1/2 cup flour onto work surface. Scrape dough out onto floured work surface (dough will be soft and sticky). Gently toss dough in flour until easy to handle. Brush away excess flour. Divide dough into 3 equal pieces. Divide 1 piece in half; shape each half into 10-inch-long log.
Arrange 1 log crosswise in each dove-shaped paper baking mold, curving ends to fit. Roll each remaining dough piece into 11-inch-long log, slightly tapered at ends. Place 1 log across dough in each mold. (If using 2 springform pans, divide dough in half; place half in each prepared pan). Cover molds (or pans) with plastic. Let stand at room temperature until dough rises to top of each mold and indentation remains when 2 fingers are pressed about 1/4 inch into dough, about 3 1/4 hours.
Glaze and Baking:.
Position rack in bottom third of oven and preheat to 375°F Finely grind sugar and whole almonds in processor. Add egg whites and almond extract; blend 10 seconds. Peel plastic off dough in molds. Spoon half of almond glaze over top of each. Sprinkle each with sliced almonds. Sift powdered sugar over. Slide rimless baking sheet under molds; slide molds directly onto oven rack.
Bake breads until brown on top and slender wooden skewer inserted into center comes out clean, about 45 minutes. Cool breads completely on rack. (Can be made ahead. Wrap; let stand at room temperature up to 2 days or freeze up to 1 week.).