Feed your starter and wait until it is peaked, domed, and floats in water. A weak starter is the single most common reason a sourdough bagel comes out dense, so do not rush this step.
Blend the cottage cheese with the milk until completely smooth, with no curd left at all. Warm the blend about 30 seconds, just enough to take the chill off. It should be barely warm, never hot, so it does not shock the starter.
Whisk the flour and salt together in the mixer bowl. Add the blended cottage cheese, the starter, the egg whites, the brown sugar, and the water. Mix on low for 1 to 2 minutes, just until everything is brought together and no dry flour is visible.
Cover and rest 30 minutes. Dairy hydrates flour much more slowly than plain water does, and this rest is what lets the flour drink properly. Skipping it is what makes people add water the dough never actually needed.
Finish mixing on low for about 5 minutes. Do not chase a windowpane, because this dough will never give you one. You are looking for a dough that has come together evenly with a little gluten development, clearing the sides of the bowl even if it still sticks at the bottom. Use a bench scraper to pull anything off the sides back into the middle. It should stretch away without immediately breaking, and it will feel stickier than you expect. Only if it is still crumbly and will not come together, add the remaining water 5g at a time.
Cover and rest 1 hour.
Do a series of stretch and folds, reaching under one side of the dough, pulling it up, and folding it over the center, then turning the bowl and repeating until you have gone all the way around. This is where any of the gluten development that gets missed in the mixer gets built, and the stickiness largely disappears as you go.
Bulk ferment at room temperature about 7 hours, until the dough is puffy and roughly 50 percent larger. With no commercial yeast this is the real ferment, so go by look and feel rather than the clock.
Divide into 6 equal pieces, about 178g each. Ball each one tightly, pressing out any large trapped air as you go. Cover and rest 15 minutes.
Poke through the center of each ball and stretch the hole to about 2 inches. Go wider than looks right, because the hole tightens while the bagels rest and closes further in the oven.
Cover and rest the shaped bagels 45 minutes to 1 hour.
Preheat the oven to 425°F. Bring the water to a boil in a wide pot, then stir in the honey and baking soda and bring it back to a gentle boil, never a hard rolling one.
Boil 30-45 seconds per side, laying each bagel in top side down first. The side that goes in first sets against undisturbed water, and the top is the face people see. Lay them in gently by hand rather than dropping them, and flip with a slotted spoon or spider.
Lift each one out, hold it over the pot a few seconds to drain, and rest 1 to 2 minutes on a cooling rack. Move to a parchment lined baking sheet.
If you are using the egg white wash, whisk the egg white with the water and brush it on now, once the surface has gone from beading wet to matte and slightly tacky. Skip the wash entirely for the most even color, since the honey in the bath gives real shine on its own.
Bake 20 to 25 minutes, rotating once, until deep golden. They are done at 195 to 205°F internal. Cool completely before slicing, because the crumb is still setting as it comes down in temperature.