TO DISSOLVE YEAST
In a large mixing bowl, dissolve 1 tsp. of sugar in warm water. Water temperature should be between 110F and 115°F.
Sprinkle yeast over top of water mixture and let stand for 10 minutes. (Yeast should be frothy – if not, wait another two to three minutes).
Whisk in eggs and vegetable oil.
Add in sugar and salt.
Gradually stir in 2 cups of the flour and continue to stir until dough is smooth. Gradually add another cup of the flour and continue to mix. The dough should be soft but not sticky.
Turn out the dough onto a very lightly floured surface. If the dough is a little sticky when you are kneading it, you can work up to ½ cup of additional flour into the dough but not more than that as it will upset the balance needed for the right texture.
Knead dough for 10 minutes. To knead dough, place dough in front of you and fold the back half of the dough towards you. Use heel of your hand to press dough down into itself. Give the dough a quarter turn and repeat. (Turn, knead, turn, knead). When kneading is completed, the dough should spring back when you poke it with your finger.
Using a different large bowl, coat the bowl lightly with olive oil.
Place dough in bowl and turn dough through the oil once to grease.
Cover with plastic wrap and let rise until doubled in size for 1 to 1 and ½ hours.
Punch down the dough and knead 10 times.
Divide dough into 12 equal portions.
In a large, wide saucepan, bring 16 cups of water and the 2 tbsp. of sugar to a boil. (you will be shaping the bagels while the water is coming to a boil).
Roll into a 10-inch long rope. Bring the end together to form a circle, overlapping, then pinch the edges to seal the dough.(Pinch the edges tightly)! Place on a parchment paper-lined baking sheet. Cover with a clean tea towel and let rise for 15 minutes.
Gently add the bagels to the boiling water, four at a time. Cook over medium heat for 1 minute. Turn over with slotted spoon and cook for 1 minute more. Move bagels to a baking sheet that is freshly-lined with parchment paper. Continue with the remaining bagels until all 12 have been completed.
Beat one egg with 1 tsp. water in a small bowl.
Brush the bagel tops with eggs, then place the bagels on parchment paper-lined baking sheet and sprinkle the cheese on top. If you do not have parchment paper, you can grease and lightly flour the baking sheets.
Bake in 400°F for 20 to 25 minutes or until tops are golden brown.
Transfer to cooling racks. See recipe notes below for storage and freezing instructions.