In a medium to large bowl, place the yeast and lukewarm water. Stir gently to mix. Allow to sit for 5 to 10 minutes until it fully dissolves. (may become foamy, which is normal)
Add 4 cups flour, sugar, salt, oil, eggs and vanilla to the yeast mixture. Stir to combine, and add flour as needed to help pull the dough together. (you can use your hands or a wooden spoon to start, but will want to switch to using your hands eventually)
Once thoroughly mixed, remove the dough from the bowl and place on a lightly floured surface. Knead the remaining ½ cup flour into the dough, continuing to knead for about 10 minutes.
Place dough in a greased bowl and cover with a light kitchen towel. Allow to rise for 2-3 hours.
Divide dough in two. Working with one half (or one loaf of dough), divide the dough further into three equal pieces. Roll out each piece into an ~18” x 4” rectangle, in preparation for stuffing these strands with the cinnamon sugar filling that is traditional for king cakes.
Spread 1 ½ - 2 tablespoons filling onto the bottom 2/3 of the rectangle of each strand. Make sure to leave ~1/2 inch border on all sides where there is no filling, so that the edges of the dough will seal nicely.
From the long edges where the filling is spread, roll up the dough to form strands. Pinch the seams closed to seal the filling inside each strand.
Braid filled strands into a three-strand challah braid, being conscious to keep the fully braided loaf long in length and shape into a circle or oval.
Place braided challah on a baking sheet lined with parchment paper or silicone baking mat. Cover with light kitchen towel and allow the challah to rise for another 45-60 minutes, or until you can see the size has grown and the challah seems light.
Preheat oven to 375°F while the dough rises.To make the egg wash, in a small bowl beat one whole egg with 1 teaspoon water. Brush the egg wash liberally over the challah. Bake for 18-20 minutes.