Dye the eggs. Dye your raw eggs and set aside to dry completely, at least 2 hours or overnight.
Make the dough. Add the warm milk, sugar, yeast, eggs, melted butter, salt, orange zest and juice, and vanilla to a stand mixer bowl. Stir to combine. Add 3 cups of flour and mix until a wet dough forms. Add remaining flour about 1/2 cup at a time until the dough pulls away from the sides of the bowl.
Knead. Using a dough hook, knead for 6-8 minutes until smooth and elastic. (Or knead by hand for 8-10 minutes.)
First rise. Place dough in a greased bowl, cover with plastic wrap, and let rise in a warm spot until doubled, about 2 hours.
Shape. Punch down the dough and divide into 3 equal pieces. Roll each into an 18-20 inch rope. Braid the ropes together, pinch the ends, and shape into a ring or straight loaf. Transfer to a parchment-lined baking sheet.
Second rise. Cover loosely and let rise 30-45 minutes until puffy.
Bake. Preheat oven to 350°F. Brush dough with beaten egg, nestle dyed eggs into the braid, and add sprinkles if using. Bake 35-40 minutes until deeply golden and internal temperature reaches 190°F. Tent with foil if browning too quickly.
Cool. Rest on baking sheet 10 minutes, then transfer to a wire rack to cool completely before slicing.