Step 1

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.

Step 2

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.

Step 3

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.

Step 4

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.

Step 5

Cover with a kitchen towel; let rise in a warm place until almost doubled, about 30 minutes. Preheat oven to 350°.

Step 6

Bake until golden brown, 30-35 minutes. Remove from pan to a wire rack to cool. Refrigerate leftovers.