Be sure to adjust the ratio of flour and water according to local climate conditions. I have found that the amount of liquid required to make a perfect dough can change slightly, just by using a new bag of flour. The dough should be soft, elastic and slightly sticky - use as little flour as possible.
By Hand: Mix together one third of the flour and all of the yeast in a large bowl. Add the sugar and salt to the warm water and pour over the flour and yeast mixture along with the egg and oil. Beat together using an electric mixer on low speed and then switch to high speed for three minutes. Add as much remaining flour as possible by mixing in with a spoon. Turn the dough out onto a lightly floured surface and knead, mixing in remaining flour to make a moderately stiff dough. Knead for at least 15-20 minutes and then cover and allow to rise for one hour.
Bread Machine: Put water, oil, and egg into the workbowl of the breadmaker and then add the dry ingredients, ending with the yeast. Set machine to 'Dough' cycle. Place dough on a lightly floured board and knead to remove large bubbles in the dough and form into a roll. Divide into ten equal portions and then roll each piece into a 14" (35 cm) length.
Lightly fold in half and twist together, then form a circle, tucking the ends into the loop. Form the bagel with light rolling and your fingers to make a hole.
Place the bagels onto a large floured baking tray, cover with a dry dish-towel and allow to rise until almost doubled in size (about one hour). Heat approximately one gallon of water, in a large pot with 2 tablespoons of brown sugar until boiling. Set aside until after the next step.
Preheat the broiler on your oven and place the baking tray about six inches (15 cm) from the element or heat source. Do not allow to brown - the purpose of this stage is to harden the outer surface of the bagel slightly. After 2-3 minutes, turn the bagels over and broil the underside lightly.
Preheat oven to 375°F or 190°C with the baking rack in the middle of the oven.
Bring the pot of water back to the boil, add the bagels, two or three at a time to the boiling water and maintain a gentle boil. Turn them with a large slotted spoon and simmer for about 2 or 3 minutes in total. Remove the bagels to a baking tray covered with parchment paper and bake in two batches for approximately 30 minutes. Do not boil the second batch of bagels until the first batch are almost ready to come out of the oven.
Optional: You can add various toppings to the bagels after boiling, brush with beaten egg and add your topping such as sesame seeds, poppy seeds, and so on, before baking.