In a large mixing bowl, combine 1 cup of flour, sugar, salt, and yeast.
Add the warm milk and butter. Mix until smooth.
Add the eggs and 2 more cups of flour. Beat until smooth.
Stir in enough remaining flour to form a soft dough.
Turn 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 and let rise in a warm place until doubled, about 1 hour.
Punch dough down. Turn onto a lightly floured surface; divide into thirds. Shape each into a 20-in. rope. Place ropes on a greased baking sheet and braid; pinch ends to seal and tuck under.
Gently split the braids apart and tuck the dyed eggs into the openings. Cover and let rise until doubled, about 30 minutes.
Bake at 350°F for 25-30 minutes or until golden brown. Remove from pan to a wire rack to cool.