Bloom the yeast first so you know it is alive before you commit the whole batch. Warm 30 grams of the water to about 110°F, stir in a pinch of the brown sugar and the active dry yeast, and let it stand 5 to 10 minutes until foamy. If it does not foam, the yeast is dead and you should start over.
Blend the cottage cheese with the milk until completely smooth, since any curd that survives the blender turns into a wet spot in the crumb. Warm the blend about 30 seconds, just enough to take the chill off.
Whisk the flour and salt together in the mixer bowl. Add the blended cottage cheese, egg whites, the rest of the brown sugar, the bloomed yeast, and the remaining 20 grams of water, then mix to a shaggy mass.
Cover and rest 15 minutes. Dairy hydrates flour slowly, so this rest lets the dough come together properly and keeps you from adding water it does not actually need.
Knead 6 to 8 minutes in the mixer on speed 2, or about 8 minutes by hand, until the dough is firm, smooth, and clears the bowl. It should feel stiff rather than soft. If it will not come together, add water a little at a time, and if it clings to the bowl, add a little flour.
Cover and bulk ferment 60 to 90 minutes at room temperature, until puffy and about half again as large. Do not let it double, because a dough taken past its peak goes slack and loses its shape in the boil.
Divide the dough into 6 equal pieces at about 172 grams each. Shape each into a tight ball, pressing out any large trapped air, then cover and rest 10 minutes.
Poke through the center of each ball and stretch the hole to about 1 3/4 inches. Open it wider than looks right, because it closes as the bagels proof and again in the oven.
Proof the shaped bagels 20 to 30 minutes. Float test one in cool water and boil the batch as soon as it floats. A poke should spring back slowly and leave only a slight indent.
Preheat the oven to 425°F and bring the water to a boil in a large pot, then stir in the honey and baking soda and bring it back to a gentle boil. A gentle boil sets a smooth, glossy skin, while a hard rolling boil sets it rough and can wrinkle the tops.
Boil the bagels 30 to 45 seconds per side, top side down first, laying them in gently and flipping with a slotted spoon. The longer end of that range gives a chewier, firmer skin.
Lift each bagel out to drain for a few seconds, rest 1 to 2 minutes on a rack, then move to the parchment lined baking sheet. If you are washing them, brush on a thin egg wash now while the surface is tacky, then add any toppings.
Bake 20 to 25 minutes, rotating once, until deep golden. They are done at 195 to 205°F in the center. The dairy browns the outside quickly, so trust the thermometer over the color, and let them cool before cutting so the crumb can finish setting.