In a large mixing bowl whisk together 2 cups of flour, sugar, yeast, and salt. Set aside
Whisk together the eggs in another small bowl and set aside.
In a small saucepan over medium heat, heat the butter and milk until melted. Remove from the heat and add water. Let mixture stand for a minute or two, or until the mixture reaches 115 to 125 degrees F
Pour the milk mixture into the dry ingredients and mix with a spatula. Add the eggs and stir until fully incorporated. Add the remaining flour and continue to stir for about 2 minutes. The mixture will be sticky.
Place the dough in a large, greased bowl. Cover with plastic wrap and allow to rest in a slightly warm area until doubled in size, about 1 hour.
While the dough rises, grease a 9x5 inch loaf pan and set aside. Combine all the ingredients for the basil filling in a blender and blend until smooth. Set aside.
Once the dough has risen, add about 2 tablespoons of flour into the dough and knead it until it's deflated. Cover with a clean kitchen towel and let rest for 5 minutes.
On a lightly floured work surface, use a rolling pin to roll the dough out to about 12-inches x 20-inches. Use a pastry brush to brush the basil vin across the dough.
Slice the dough vertically, into six equal-sized strips. Stack the strips on top of one another and slice the stack into four equal slices once again. You’ll have six stacks of 4 squares. Stack all the squares on top of each other and set into the prepared loaf pan. Place a kitchen towel over the loaf pan and place it in a warm area for 30 to 45 minutes until almost doubled in size.
Preheat your oven to 350 degrees F. Place the loaf pan into the oven and bake for 30 to 35 minutes, until the top is very golden brown. The top may be lightly browned, but the center may still be raw. A nice, dark, golden brown will ensure that the center is cooked as well. Allow the loaf to rest for about 20 minutes and then run a knife around the edges of the pan to loosen and carefully turn the loaf out, transferring to a serving plate. Serve warm with extra basil vinaigrette brushed on top if needed