Combine milk, oil, salt, orange extract, 2 eggs at room temperature, sugar, orange zest, 4 cups of flour, and yeast in a bread machine, in that order. Set to Dough cycle and start the machine; be sure to peek inside while dough is mixing and add additional flour as needed to create a soft, but not wet, dough.

Transfer dough to an oiled bowl and let rise until doubled in size, about 1 hour.

Meanwhile, bring a pot of water to a boil over medium-high heat. Reduce heat to a rapid simmer and add 4 whole eggs to soft boil. Continue to simmer for 6 minutes. Remove from heat and transfer eggs to a bowl of ice water to cool quickly.

Punch risen dough down and turn onto a floured surface. Divide into 3 even pieces. Roll each piece to form about a 1 1/2-inch thick rope that is 12 inches long. Take the pieces and braid them together, pinching the ends. Loop into a circle and pinch the ends to close. Place on a baking sheet lined with parchment paper. Align boiled eggs evenly about the braid and push in gently, about 1/3 of the way down. Cover and let rise again, about 1 hour.

Combine remaining room temperature egg, 2 tablespoons water, and anise extract in a bowl for egg wash. Brush braid with the egg wash, being careful not to get any on your boiled eggs or while baking they will have cooked wash on them and won't dye nicely.

Bake in the preheated oven, switching between another parchment-lined baking sheet every 10 minutes throughout baking time to prevent the bottom from getting too dark, until browned, about 30 minutes. Remove from the oven and let cool thoroughly, 20 to 30 minutes.

Paint your eggs on the bread with liquid food dye. Be careful not to drip on the bread. Let dry, about 10 minutes. Brush bread with anisette liqueur and let dry before serving, about 5 minutes more.