Combine yeast, sugar, and warm water and let sit until bubbly.
Make the flax egg by combining flax meal and water.
In a mixer with a dough hook or food processor with the bread blade, combine yeast mixture, flax egg, winter squash, and water. Mix to blend.
Gradually add 5 cups of bread flour. Add a little more flour if the dough is still sticky and knead about 8 minutes to develop the gluten.
Place dough into a bowl, cover with a light towel, and place in a warm spot to rise 1 hour. I simply turn on the oven light (with oven turned off) and put the covered bowl in there—it's the perfect proofing temperature.
After dough has risen, remove from the oven, punch down, and cut into 12 100g pieces. Cover pieces with a towel while shaping individual bagel so the dough doesn't dry out while you are working.
Roll piece of dough into a log shape and then wrap it around your hand. Pinch the ends together well and roll the O-shaped dough with your other hand or on the counter, using your wrapped hand as a form.
Place the formed bagel on a cookie sheet lined with lightly floured cloth. Repeat with remaining dough.
Cover formed bagel and let rise 1 hour.
Heat oven to 375°F. If using a baking stone, preheat oven with stone.
Bring a pot of water to boil. Add salt to taste once water is boiling, it should taste like seawater.
Add 3 bagel at a time to the boiling salt water. Boil 30 seconds, then flip and boil another 30 seconds. Remove boiled bagel to baking sheet covered with a silicone baking sheet. Repeat with remaining bagel.
If you want sprinkles like poppy seeds, sesame seeds, onion flakes, etc., add them now so they stick to the top.
Bake 30 minutes, either removing prepared bagel to a preheated baking stone or on the baking sheet.