Activate the Yeast:
In the bowl of a stand mixer, combine 1/3 cup hot water (about 110°F) with the yeast, 1 tbsp of flour, and 1 tbsp of sugar. Whisk gently, cover with a towel, and leave aside for 10 minutes until bubbly.

Prepare Milk Mixture:
Heat the milk in a saucepan over medium heat until the edges begin to bubble (about 4–6 minutes). Remove from heat and stir in butter, remaining sugar, and salt. Let it cool for 5 minutes.

Mix the Dough:
Add the milk mixture into the yeast mixture in the stand mixer. While stirring with the paddle attachment, gradually add the 2 beaten eggs. Then, mix in flour, anise seed, and vanilla extract. Beat until a soft dough forms. Increase speed to medium-high until the dough pulls away from the sides of the bowl.

Knead the Dough:
Transfer the dough to a lightly floured surface (it will be sticky). Knead by hand for about 5 minutes until smooth. Place the dough into a lightly oiled bowl, cover, and let it rise for 1 1/2 hours in a warm, closed space until doubled in size.

Shape the Loaf:
Punch down the dough, transfer it to a floured surface, and divide it into 3 equal portions. Roll each portion into a long 16-inch rope. Place the ropes on a parchment-lined baking sheet and braid them loosely. Form the braided dough into a circle and pinch the ends together. Stretch the circle to about 10 inches across with a 3 1/2-inch wide hole in the center.

Second Rise:
Cover the loaf with a clean towel and let it rise for 1 hour in a warm space until doubled in size.

Prepare the Eggs:
Rub the dyed hard-boiled eggs with olive oil and buff with a paper towel. Gently insert the eggs into three evenly spaced spots in the dough. Cover and return to the warm space for 1 more hour until the dough rises again.

Bake:
Preheat the oven to 350°F (175°C). Brush the loaf with the egg wash and sprinkle with toasted sesame seeds, if desired. Bake for 45 minutes, or until the loaf turns golden brown and sounds hollow when tapped.

Cool and Serve:
Let the bread cool for about 30 minutes before serving. Enjoy with additional boiled eggs, honey, or other breakfast items!