In a large bowl, mix sugar, yeast, cardamom, salt and 2 cups flour. In a small saucepan, heat milk and butter to 120°-130°. Add to dry ingredients; beat on medium speed 2 minutes. Add 3 eggs; beat 2 minutes longer. Stir in enough remaining flour to form a soft dough (dough will be sticky).
Turn dough onto a floured surface; knead until smooth and elastic, about 6-8 minutes. Place in a greased bowl, turning once to grease the top. Cover with plastic wrap and let rise in a warm place until doubled, about 45 minutes.
Meanwhile, color hard-cooked eggs with food coloring following package directions; let stand until completely dry. Punch down dough. Turn onto a lightly floured surface; divide into thirds. Roll each portion into a 24-in. rope. Place ropes on a greased baking sheet and braid. Bring ends together to form a ring. Pinch ends to seal. Lightly coat dyed eggs with oil; place on braid, tucking eggs between ropes and spacing evenly.
Cover with a kitchen towel; let rise in a warm place until doubled, about 20 minutes. Preheat oven to 375°.
In a bowl, whisk remaining egg and water; gently brush over dough, avoiding eggs. Bake 28-32 or until golden brown. Remove from pan to a wire rack to cool. Refrigerate leftovers. Yield: 1 loaf (16 slices).