Prepare the Dough:
In a small bowl, place the yeast, half teaspoon sugar, and lukewarm water. Stir gently to mix. Allow to sit five to 10 minutes, until it becomes foamy on top.
In a large bowl or stand mixer fitted with the whisk attachment, mix together one and a half cups of the flour, salt, and the remainder of the sugar. Add the water-yeast mixture and oil to flour. Mix thoroughly.
Add another cup of the flour and two eggs and mix until smooth. Switch to the dough hook attachment if you are using a stand mixer.
Add another one and a half to two cups of the flour and mix thoroughly. Remove from the bowl and place on a floured surface. Knead the remaining half cup flour into the dough, continuing to knead for about five minutes.
Place dough in a greased bowl and cover with a damp towel. Allow to rise at least three hours, punching down at least once if possible.
Shape and Bake:
When fully risen, split the dough into two parts.
To make a braided stuffed challah (see note for a variant): Split each dough section into three additional sections. Roll each into a rope about six inches long. Flatten the ropes.
Spread one-third of the pesto in the middle of each rope. Top each with one-third of the cheese. It may seem like a lot as you are doing it, and it will be messy, but the amount of filling can be deceiving.
Fold the sides of each rope up over the filling and pinch tightly. Roll slightly to even out the shape. Braid.
Working quickly, roll each rectangle from the longer end into a long rope. Try to keep the roll relatively tight as you go. Pinch the ends when you finish. Circle the dough around itself and then pinch under.
Place challahs on a baking sheet lined with Parchment Paper or a silicone baking mat. Allow challah to rise another 30 to 45 minutes, or until you can see the size has grown and dough seems light. Preheat oven to 375 degrees while the dough rises.
In a small bowl beat two egg yolks with one teaspoon water.
Brush the challah liberally with egg wash and top with dried basil and coarse sea salt. Bake for 24 to 26 minutes, or until golden brown.