Add milk, honey, vanilla, and salt to the bowl of a stand mixer and whisk together until combined.
Add in 4 cups (520 grams) of flour, mixing until the dough comes together.
Let sit for 15 minutes - the dough will be crumbly.
Add in eggs.
Cut softened butter into 4 pieces and mix one piece at a time into the dough, about 1 minute each, until each one is fully incorporated. The dough will be sticky.
After adding in the butter, add in the remaining ½ cup (63 grams) of flour and the yeast. The dough should come together better, but will still be somewhat sticky.
Knead for 8-15 minutes until the dough is smooth and shiny.
Form dough into a ball and place in a greased bowl.
Let rise for 1 hour in a warm place until slightly risen.
After it has risen a little, punch the dough down, form into a ball again, and refrigerate overnight (10-24 hours).
Remove dough from the fridge and allow to warm up for about 3 hours.
Once the dough is room temperature, dump out onto an oiled surface.
Cut the dough in half and cut each half into 3 equal pieces.
Roll up each of the 3 pieces into a log, about 12-14 inches long and 1 inch in diameter.
On one side, connect the 3 ends together by pinching them together.
Next, braid the pieces together, pinching the ends together once you get to the other end.
Repeat this process with the other 3 pieces.
You can either place the braided dough into two 9 x 5 inch greased loaf pans or you can place them on a baking sheet lined with parchment paper (or one of each).
Allow to rise in a warm place, covered, for about an hour. They should be pretty well risen, but don't expect them to fully double in size.
Preheat oven to 325 degrees F.
Whisk one egg with 1 teaspoon of water and brush over both loaves.
Bake for 32-38 minutes until golden.
Optional: brush with melted butter.