In a mixing bowl (of your stand mixer), combine the flour, yeast, sugar and salt.
In another small bowl, whisk together the water, milk and egg.
With the paddle attachment in your stand mixer or a dough whisk, mix the dry ingredients and add the water-milk-egg mixture and mix to combine.
Switch to the dough hook of your stand mixer, and knead the dough to bring it together. Add more water if the dough is too dry. Knead for about 5 minutes, until you have a smooth ball of dough.
Add the oil or butter and knead for a further 5-10 minutes (I knead it for about 8 minutes), until the dough is shiny, smooth and a little tacky to the touch.
Lightly oil a big bowl and place the dough in it. Make sure the dough is lightly oiled on the surface as well.
For best results, cover the bowl with plastic wrap and store in the fridge overnight (8 hours) or for up to 24 hours. Alternatively, if you want to bake it on the same day, you can cover the dough with plastic wrap and let it rise for 1 hour in a warm place, until it has doubled in size.
On the day you bake the bread rolls - turn the chilled dough out on to a lightly floured surface and lightly dust the surface of the dough as well. If the dough wasn't chilled, you may need extra flour on your hands and on your work surface to prevent the dough from sticking.
Roll out the dough into a rectangle that's about 16 inches in length and 4-5 inches in width. Cut the dough into 10 sections and place them on a parchment paper lined baking tray, with about 1.5 inches of space between each.
Cut the dough into 12 (or 10) equal sections. I usually weigh the bread dough so that I know how much each section should weigh.
Knead each portion for just a few seconds to form a smooth ball. Roll out each portion into a flattened piece of dough that's about 5 inches long and 4 inches wide. Then TIGHTLY roll up the dough to form a cylinder. Tuck in the sides/edges and pinch these edges and seams to seal. Place each portion on a parchment paper lined baking tray, about 2 inches apart.
Cut the dough into 8 sections (for larger buns) or 10 sections (for smaller buns) and roll these portions in your hands to form smooth balls. Place them on the parchment paper lined baking tray, about 3 inches apart.
Cover the dough with plastic wrap and let it rise for 45 min in a warm place, until it has just about doubled in size. In winter, this can take a little longer than 45 min, and on very hot summer days, it can take less than 45 minutes.
Brush the hot dog rolls with the egg wash (lightly beaten egg) and bake in the preheated oven until they are golden brown on top and sounds hollow when tapped, which should take about 15-20 mins. (Optional - sprinkle sesame seeds or poppy seeds on the egg wash brushed dough before baking). SEE NOTES FOR FREEZING INSTRUCTIONS.
Remove from the oven and let the buns cool down enough to be handled comfortably. Best eaten on the same day. Will keep longer in an air-tight container in the fridge.
NOTES on freezing - bake the hot dog buns until they are just starting to turn golden brown - about 10 to 15 minutes. Remove the buns from the oven, and allow them to cool down. FREEZE the hot dog buns at this point, and then place them in an air-tight container (or freezer bag).
When you're ready to bake, allow the buns to thaw out in the fridge first. Preheat oven to 350°F and place the thawed out hot dog buns on the tray and bake for about 10 to 15 minutes until the buns are golden brown on top.