Pour 1/4 cup of warm water into the bowl of an electric mixer and sprinkle the yeast on top. Let sit for 5 minutes for the yeast to develop. Add the remaining 2 cups warm water, sugar, vegetable shortening and salt and mix on low speed with a dough hook attachment.
Gradually add 5 1/2 cups of bread flour, mixing for 2-3 minutes until all ingredients are combined. Increase to medium speed and mix for 6 minutes, adding the remaining flour 1 tablespoon at a time.
Brush the inside of a large bowl with some of the melted butter. Shape the dough into a ball and place inside the bowl. Brush the top of the bun with some more butter and cover with a sheet of buttered plastic wrap. Place a towel on top and let the dough rise at room temperature for 1 hour.
Transfer the dough to a floured surface and dye it pink with food colouring. It will take a few minutes for the food colouring to evenly disperse throughout the dough, but keep kneading and you will eventually have a smooth, even colour.
Place the dough in a buttered bowl and brush some more butter on top. Cover with a buttered sheet of plastic wrap and a dish towel and let rise at room temperature for 1 hour 30 minutes. Then transfer the bowl to the fridge and chill for 1 hour.
In the meantime, do some prep. Spray 2 baking sheets with cooking spray and sprinkle sesame seeds on top. Cover 2 additional baking sheets with dish towels and sprinkle flour onto one of the dish towels. Set aside.
Transfer the dough to a floured surface. Deflate the dough and divide into 10 pieces. Roll them into sausage shapes, then shape them into rings by pinching the ends together. Transfer the bagels to the baking sheet with the floured dish towel and cover with an additional dish towel.
Bring a large pot of water to boil and add the sugar and baking soda. Place the bagels into the boiling water and boil for 1 1/2 - 2 minutes, then flip over and boil for an additional 1 1/2 - 2 minutes on the other side. Transfer the boiled bagels to the baking sheet lined with the un-floured dish towel. Then transfer the bagels to one of the baking sheets sprinkled with sesame seeds.
Preheat the oven to 500F. While the oven is preheating, beat the egg whites and water in a small bowl. Brush the bagels with this glaze, and if desired, sprinkle with additional sesame seeds. Place a sheet of aluminum foil over the bagels (this will prevent them from browning and will maintain the bright pink colour).
In a small glass, place 4 ice cubes and 1/4 cup water.
Place the bagels in the oven and immediately toss the ice cubes and water onto the over floor. Quickly close the oven door and reduce the oven temperature to 450F. Bake for 25 minutes. Turn off the oven and keep the bagels in the oven for an additional 5 minutes. Then open the oven door and leave the bagels in the oven for another 5 minutes. While the first batch is baking, prep the other bagels using the remaining dough.
After cooking, transfer the bagels to a cooling rack and enjoy!