In a stand mixer fitted with a dough hook, combine the orange juice, 1 tablespoon water, and yeast. Mix until frothy.
Add the olive oil, 2 eggs, 1 egg yolk, honey, sugar, and salt. Mix until combined.
Add the flour one cup at a time, adding the extra tablespoon with the final cup, until the dough is slightly sticky and begins to pull away from the sides of the bowl.
Transfer the dough to a floured surface and knead until smooth.
Place the dough into an olive oil–lined bowl. Cover with a tea towel and let rise for 1½ hours.
After the first rise, punch down the dough. Cover and let rise again for 45 minutes.
In a small bowl, combine 1 egg and 1 teaspoon water to make an egg wash.
Divide the dough into 3 equal sections and roll each into a 12-inch rope.
Braid the ropes, tuck the ends under the loaf, and brush with egg wash.