How to Make It
Place 1/2 cup hot water in each of 6 small bowls or cups. Stir 1 tablespoon vinegar and desired amount of food coloring gel into hot water in each bowl until blended. Place 1 egg (in shell) in warm dye in each bowl, and let stand until entire shell is dyed, about 1 1/2 minutes per side. Transfer eggs to paper towels, and let stand until completely dry, about 1 hour.
Meanwhile, place warm milk, sugar, butter, instant yeast, and 2 of the eggs in bowl of stand mixer fitted with a paddle attachment. Beat on medium speed until combined, about 1 minute. Gradually add flour and salt, beating on low speed until a dough forms, about 3 minutes. Fit mixer with dough hook, and beat on medium speed until dough is elastic and pulls away from sides of bowl (dough will stick to bottom of bowl), about 5 minutes. Transfer dough to a large bowl greased with cooking spray, cover with plastic wrap, and let rise in a warm place (80°F to 85°F) , free from drafts, until doubled in size, about 1 1/2 hours.
On a lightly floured surface, punch dough down, and divide into 3 equal portions. Roll each portion into a 28-inch-long rope. Braid ropes together, pinching ends to seal. Shape braided dough into a round wreath shape, pinching ends together to seal. Transfer to a large parchment paper-lined baking sheet. Tuck dyed eggs around wreath into braided dough. Cover with plastic wrap, and let stand until almost doubled in size, 45 to 60 minutes.
Preheat oven to 350°F. Whisk remaining egg in a small bowl. Uncover dough wreath, and brush with egg wash. Bake in preheated oven until golden brown and cooked through, about 28 minutes. Carefully transfer wreath on parchment paper to a wire rack, and let cool completely, about 45 minutes.
Whisk together powdered sugar, orange zest, orange juice, and cream in a medium bowl until smooth. Drizzle over bread, and top with sprinkles.