Easter Egg Bread

Bring vinegar and 10 cups of the water to a boil in a large saucepan over high. Carefully divide hot water mixture evenly among 5 bowls, and add food coloring gel to each bowl to achieve desired colors. Place 1 of the hard-cooked eggs in each bowl, and let stand until eggs are dyed, about 5 minutes. Remove dyed eggs from water mixture, and transfer to a rimmed baking sheet lined with paper towels; refrigerate, uncovered, for at least 1 hour or up to 12 hours.

Meanwhile, heat milk and butter in a small saucepan over medium-low, stirring occasionally, until butter is completely melted and a thermometer inserted in milk mixture registers 120°F, about 5 minutes. Beat sugar, yeast, cardamom, salt, and 2 cups of the flour with a stand mixer fitted with a dough hook attachment on low speed until well combined, about 30 seconds. Add warm milk mixture to sugar mixture, and beat on medium speed until combined and very smooth, about 2 minutes. Add 3 of the large eggs to flour mixture, and beat until combined, about 2 minutes. Add remaining 4 1/2 cups flour, and beat until well combined and dough is sticky, about 2 minutes. Transfer dough to a work surface lightly dusted with flour, and knead until dough becomes smooth and elastic, about 6 minutes, adding more flour to work surface as needed to prevent dough from sticking. Transfer dough to a large bowl lightly greased with cooking spray, cover with a kitchen towel or wrap tightly with plastic wrap, and allow dough to rise in a warm place (80°F to 85°F), free from drafts, until doubled in size, about 45 minutes to 1 hour.

Line a rimmed baking sheet with parchment paper; set aside. Uncover dough, and punch down to remove air pockets. Transfer dough to a work surface lightly dusted with flour, and divide dough into thirds. Roll each third into a 24-inch-long rope, and transfer to prepared sheet pan. Gently braid dough pieces tightly together, and shape into a ring; pinch dough ends together to seal. Remove dyed eggs from refrigerator, and brush with oil. Arrange dyed eggs on braided dough ring, gently tucking eggs into braided dough. Cover dough with a kitchen towel or plastic wrap, and let rise in a warm place  until doubled in size, about 20 to 30 minutes. Preheat oven to 375°F. Whisk together remaining 1 egg and remaining 2 tablespoons water in a small bowl, and gently brush mixture over dough, being careful not to brush eggs. Bake dough in preheated oven until golden brown and fragrant, about 25 to 30 minutes. Let bread cool on baking sheet 5 minutes; transfer to a wire rack to cool until ready to serve. Slice bread, and serve warm (eggs can be peeled and eaten, if desired, or discarded); cover and refrigerate leftovers for up to 3 days.