All my egg white increases have been in quick breads, so that isn't helpful. But – in case your goal is adding protein to your bread without negatively impacting the texture: using the whey from homemade cheese is a flavorful replacement for water in yeast breads that has not resulted in dryness, flatness, or scorch risk for me.
I don't recall the impact on crumb.
Don't expect egg whites to strengthen the gluten structure. Any old protein won't form a springy dough that can handle air bubbles the way gluten can. Gluten is special because it has long strands that can line up and link together.
This is why dough gets tight and stiff as you knead it; all those gluten strands are being lined up.
I'm not sure if your goal is using egg whites or getting bigger air bubbles… if you just want to use egg whites, make a soft enriched bread. If you want a nice chewy loaf with an open crumb, make a high hydration lean dough.
I don't have a scale so it is a little bit more about eying it. My challah recipe, which I use on a regular basis:
- Tbsp dry yeast
- 1/2 cup sugar
- 1 1/2 cup warm water
- 1 tsp salt
- 1 egg white (set aside the yolk for the wash)
- 1/2 cup oil
- 5 1/2 cups flour (wheat or white)
- 6 cups fir high altitude
- 1/4 cup gluten if using a high percentage of wheat
(I make this using the bread machine dough cycle) Combine water yeast and sugar. Mix and let stand for 10 minutes until it is very foamy. Add the rest of the ingredients in order, mixing well after each one. Once the dough is formed, knead for 5 minutes. Cover and let rise until doubled. Heat oven to 350 F. Punch down and shape as desired. Cover and let rise for 10-15 minutes Mix the yolk with a little water and paint it gently with a brush. Sprinkle with seeds (poppy ,sesame, sunflower, etc) Bake in preheated oven until bottom is hard when you tap on it.(about 25 min.) Remove, place on rack to cool for at least 15 min.