For the egg wash, keep aside 1 egg white whisked with 1 teaspoon of water. Before you begin, do ensure that water and eggs are at room temperature.
Stir together flour, sugar, salt and yeast in a large mixing bowl.
In a separate bowl whisk together the oil, eggs and yolks and the water till frothy.
Pour the egg mixture in the flour mixture.
Mix with a spoon until all the ingredients gather and form a ball. Add more water spoon by spoon only if needed to make a course tacky dough.
Sprinkle flour on the counter, transfer the dough to the counter, knead for 10 minutes sprinkling in flour or water if needed to make the dough soft and supple, slightly tacky but not sticky.
Dough should be stretchy and register approximately 77deg F to 81 degrees F.
Lightly oil a large bowl. Form the dough into a ball and transfer it to the bowl, rolling it around to coat with oil. Cover with plastic wrap. Ferment for 1 hour at room temperature.
Remove dough and punch it to de gas. Knead for 2 mins, form a ball, return to the bowl and allow to rise till 1.5 times it's original size (around 1 hour and 20 minutes in Mumbai weather).
Remove dough from bowl and divide into two equal pieces. Form them into balls and let them rest for 10mins on the counter covered with a damp towel.
Roll out each ball with a rolling pin, should be atleast 1/2 inch thick. Remove the khaman from the fridge and put the filling in the centre. Leaving two inches from the top and the bottom. To seal pinch the dough shut tightly, turn the seal portion on the counter and roll gently with palms to make sure the dough spreads evenly. Repeat with the second ball of dough.
Once the strands are ready pinch both of them together at the top and start twisting them carefully one over the other making sure that the khaman doesn't ooze out.
When you get to the end of the braid pinch them together again. Place them on a lined sheet pan with baking parchment. Brush with egg wash. Mist them with spray oil, cover loosely with plastic wrap. Proof at room temp for 1 to 1 1/4 hours until the dough has risen to 1.5 times its original size.
Preheat oven to 350deg F, with the rack on the middle shelf. Brush with egg wash again and put in oven for 20 mins. Rotate the pan and continue baking for 20/45 mins. The bread should have a brown/ golden hue and register 190 deg F in the centre.
Transfer to a rack and let cool for atleast 1 hour before slicing.