Place 3 eggs in a single layer in a small saucepan; add enough cold water to cover by 1 in. Cover and quickly bring to a boil. Remove from heat. Let stand 15 minutes.
Rinse eggs in cold water and place in ice water until completely cooled. Drain; dye hard-boiled eggs with food coloring, following package directions. Let stand until completely dry.
In bread machine pan, place the first 7 bread ingredients in order suggested by manufacturer. Select dough setting. Check dough after 5 minutes of mixing; add 1-2 tablespoons additional milk or flour if needed.
When cycle is completed, turn dough onto a lightly floured surface. Punch down dough; divide into thirds. Roll each into a 15-in. rope. Place ropes on a greased baking sheet and braid. Shape into a ring. Pinch ends to seal. Lightly coat dyed eggs with oil; arrange on braid, tucking them carefully between ropes. For egg wash, whisk egg with water. Brush over dough; sprinkle with sesame seeds.
Cover with a kitchen towel; let rise in a warm place until almost doubled, about 30 minutes. Preheat oven to 350°.
Bake until golden brown, 30-35 minutes. Remove from pan to a wire rack to cool. Refrigerate leftovers.
To prepare dough by hand: In a large bowl, mix sugar, yeast, salt and 1 cup flour. In a small saucepan, heat milk and butter to 120°-130°. Add to dry ingredients; beat on medium speed 2 minutes. Add eggs; beat on high 2 minutes. 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 rest 1 hour. Shape and bake as directed.