Dye 5 raw eggs using a store-bought kit with cold or room-temperature water. Let them dry completely before using.
Warm milk to 110°F (43°C), then combine with sugar and yeast. Let rest for 5-10 minutes until foamy.
Add the 3 room-temperature eggs, vanilla, orange zest, softened butter, and salt to the yeast mixture. Gradually add flour, mixing until a shaggy ball forms.
Knead the dough with a dough hook on medium speed for 8-10 minutes until smooth, elastic, and slightly tacky.
Place the dough in a greased bowl, cover, and let rise in a warm place until doubled in size, about 1 to 1.5 hours.
Punch down the dough and divide it into 6 equal pieces. Roll each piece into a 16-18 inch rope.
Take 3 ropes, pinch the ends together, and braid loosely. Form into a ring by bringing the sealed ends together. Repeat with the remaining 3 ropes.
Place rings on a parchment-lined baking sheet. Gently tuck dyed eggs into the gaps of each braid.
Cover and let rise again until puffed, about 45 minutes to 1 hour. Meanwhile, preheat oven to 350°F (175°C).
Whisk together the beaten egg and 1 tablespoon of milk. Brush the egg wash over the exposed dough, avoiding the dyed eggs.
Bake for 25-30 minutes until deep golden brown and the internal temperature reaches 190°F (88°C). Tent with foil if browning too quickly.
Let cool on the baking sheet for 10 minutes, then transfer to a wire cooling rack to cool completely.
Whisk together powdered sugar, 2-3 tablespoons of milk, and vanilla until smooth. Drizzle over the cooled bread and immediately top with sprinkles.
Allow the glaze to set for 10-15 minutes before slicing and serving.