The red Easter eggs that make this traditional bread so special are for decorative purposes only. Do not eat them.
Combine 1 1/2 cups all-purpose flour, sugar, and yeast in the bowl of a stand mixer fitted with whisk attachment; whisk on low speed to combine. Heat milk in a small saucepan over medium heat until it registers 110 degrees. With mixer speed on medium-low, gradually add milk to mixer bowl, scraping down sides if necessary. Mix until combined. Remove bowl, cover with plastic wrap, and let stand until bubbling, about 30 minutes.
Generously butter a large bowl; set aside. Return mixer bowl to stand and attach dough hook. Mix in lightly beaten eggs, in 3 additions, on medium speed. Add remaining all-purpose flour and salt and mix until incorporated. Add butter, orange zest and juice, and vanilla, mixing until combined. Gradually mix in 2 cups bread flour, scraping down sides of bowl if necessary; then add enough remaining bread flour to form a soft, slightly sticky dough that pulls away from the side of bowl. Increase speed to medium-high and mix until dough is very smooth, about 8 minutes. Form dough into a ball, place in buttered bowl, and turn to coat. Cover tightly with plastic wrap and let rise in a warm place until 1 1/2 times its original size, about 1 1/2 hours.
Line two 11-by-17-inch rimmed baking sheets with parchment. Punch down dough. Cut dough in half with a large knife or bench scraper, then cut each half into 3 equal pieces. Keeping 3 pieces covered with plastic wrap, roll remaining 3 pieces into 16-inch-long ropes with your hands. (If dough becomes dry, lightly dampen your palms to help in rolling.) Loosely braid ropes together, pinch ends to seal, and place loaf diagonally on 1 baking sheet. Repeat process with remaining 3 pieces of dough. Loosely cover each loaf with plastic wrap coated with cooking spray and let rise in a warm place until doubled in size, 1 to 1 1/2 hours.
Preheat oven to 400 degrees with racks in upper and lower thirds. Transfer baking sheets to oven and bake bread 10 minutes. Remove 1 baking sheet from oven. Quickly and carefully push 5 eggs, larger side down, into the center of the loaf, spacing them evenly down the length. Brush with egg wash and immediately return to oven. Repeat with remaining loaf. Reduce oven temperature to 350 degrees and bake, rotating baking sheets from top to bottom and front to back halfway through, until loaves are deep golden and sound hollow when tapped on bottom, about 25 minutes more. (If necessary, cover with parchment-lined foil to prevent overbrowning.) Transfer bread to a wire rack; let cool completely.