In a small bowl, combine the water, yeast and 1 teaspoon of sugar. Leave the mixture to stand for 3 minutes and then stir until the yeast is dissolved. Then leave it for a further 5 minutes.
Combine 4 1/2 cups of flour with the salt and remaining sugar in a large, deep mixing bowl. Add the yeast mixture, then add the eggs and vegetable oil. Gently stir until the ingredients are blended together. Stirring more vigorously, blend in the remaining flour. Turn the dough out onto a lightly floured surface and knead for 20 minutes.
Place the dough ball into a bowl, cover the bowl with a clean cloth, and let it stand for 45 minutes or until the dough has nearly doubled in bulk. Turn the dough out and knead for 5 minutes. Let it rest for 5 another minutes.
Cut the dough in half with a sharp knife, then cut each half into thirds. Roll each piece of dough into a long cylinder that tapers at each end, somewhat cigar shaped. Pinch the ends of the 3 long dough pieces together at one end. Braid the 3 pieces, then pinch the other ends together. Repeat with the remaining dough. (Or cut the dough into 4 pieces and braid them. The images shows a 4 piece braid).
Place the each dough braid on a lightly greased cookie sheet and leave them to stand for 30 minutes.
Preheat the oven to 400°F.
In a small bowl, beat the egg yolk with 2 tablespoons of cold water. Brush the tops of the challahs with the egg yolk and bake for 20 minutes. Reduce the temperature to 375°F and bake for 30 minutes more, or until the challah bread is a nice golden brown and sounds hollow when you tap it. Cool on a rack before cutting.