In a large mixing bowl, combine the yeast, warm water, sugar and salt. Let it rest for 10 minutes until the yeast is activated. You can tell its ready when the water surface is covered with bubbles. If you are using instant yeast, then you can combine and go to the next step.
Add the Egg and both Oils, if you don't have olive oil, double the vegetable oil amount. Now add the flour and knead in a stand mixer for 7-8 minutes.
Line the inside of a large bowl with vegetable oil, about 1 table spoon should suffice. Place the dough in and give it a roll around to ensure e it is covered with the oil. (this will avoid sticking to the sides and drying out) Let the dough rest for 15 minutes.
Now cut in to 12 equal portions, 6 if you are making hamburger buns. Roll the bun in to a ball. To get a nice ball shape, take a dough piece and hold with both hands. Now using your thumb, stretch the top of the dough over the edge and use your index and middle finger to tuck the under side in until you have a round ball. Pinch the bottom together and place on your working surface. Repeat with all pieces.
Extra Step (Optional) If you want perfectly round buns, use this technique. Place the rounder dough ball on your working surface (no flour needed). Form a claw over the ball with your hand and apply some pressure with your palm. Rub in a circular motion slowly releasing pressure until you have a perfectly round bun,
If you are making pull-apart bread, pack the dough balls in to a baking dish. They will grow in size and stick together, but that is the plan :)
If you are making individual buns, line a baking tray with baking paper, and place four to six buns on a tray according to your tray size. You will need 1-2 inches of space between each bun.
Cover and let the dough balls rest for 15 minutes.
Now which together the egg and water and brush the dinner rolls evenly. Place in the oven and bake at 170C or 340F for around 20 minutes or until golden dark brown.