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Whisk together 3 cups of flour, sugar, yeast, and salt in medium bowl; set aside. Mix together 2 eggs, egg yolk, melted butter, and 1/2 cup of water in bowl of standing mixer fitted with dough hook. Add flour mixture to wet mixture; knead at low speed until dough ball forms, about 5 minutes, adding remaining 1/4 cup flour, 1 tablespoon at a time, as needed to prevent dough from sticking. Whisk reserved egg white with remaining 1 tablespoon water in small bowl, cover with plastic wrap, and refrigerate.
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2. Transfer dough to very lightly oiled large bowl, turning dough over to coat with oil. Cover with plastic wrap and let rise in warm place until doubled in size, 1 1/2 to 2 hours. Gently press dough to deflate, cover with plastic wrap, and let rise until doubled in size again, 40 to 60 minutes.
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3. Lightly grease large baking sheet and set aside. Transfer dough to lightly floured work surface. Divide dough into 2 pieces, one roughly half size of other. (Small piece will weigh about 9 ounces, larger piece about 18 ounces.) Divide large piece into 3 equal pieces. Roll each piece into 16-inch-long rope, about 1 inch in diameter. Line up ropes of dough side by side and pinch ends together. Take dough rope on bottom and lay it over center rope. Take dough rope on top and lay it over center rope. Repeat until ropes of dough are entirely braided, then pinch ends together. Place braid on baking sheet. Divide smaller piece of dough into 3 equal pieces. Roll each piece into 16-inch-long rope, about 1/2 inch in diameter. Braid together, pinching ends to seal. Brush some of egg wash on top of large loaf and place small braid on larger braid. Loosely drape loaf with plastic wrap and let rise in warm place until loaf becomes puffy and increases in size by a third, 30 to 45 minutes.
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4. Adjust oven rack to lower-middle position and heat oven to 375 degrees. Brush loaf with remaining egg wash and sprinkle with poppy or sesame seeds, if using. Bake until loaf is golden brown and instant-read thermometer inserted into side of loaf reads 190 degrees, 30 to 40 minutes. Place baking sheet on wire rack. Cool loaf completely before slicing.
