Bread – Why is the bread dough sticky

While it might seem like a drastic step to double the amount of egg in your recipe, in fact, you are not adding that much more egg relative to the amounts of your other ingredients.

A whole large egg weighs about 50 grams and is roughly 75% water so that extra half egg will contribute just under 20 grams of additional liquid to your recipe - you will need to compensate for that additional liquid either by reducing your water contribution accordingly or by adding a little extra flour (like 30-ish grams).

That means you are only adding an additional 5 grams of protein and fat from the egg by including that extra half. You will probably notice a slightly more tender or soft texture in your finished loaf from the slight increase in egg.

Since texture and flavor are so subjective and difficult to describe, the best way I can think of to communicate the possible change is to ask you to think about Challah bread. Challah bread owes much of its characteristic texture, color, and flavor to the inclusion of a lot of egg in the dough. While adding that extra half egg won't bring your recipe into Challah territory, it will advance your bread's texture, color, and flavor along the spectrum in the Challah direction.

Your recipe is to blame.

Your dough is stiff and tears because it isn't wet enough. While different types of flour do absorb different quantities of water, I think most of the issues you're experiencing are due to the ambiguousness of the recipe and instructions you're using. Most sourdough recipes yield dough around 80% hydration (the weight of water/the weight of flour). The recipe you've cited relies on the starter providing all the hydration for the dough. Doing the math, following the feeding instructions, you would have a starter that is 55% hydration. Adding all the flour called for in the dough recipe, you would end up with a dough that is about 45% hydration... stiffer than bagels (50-55%).

While it is possible to make dough that doesn't have any added water, it is not common, especially with such a stiff starter. Also, the volume of flour would be much smaller than the volume of starter.

Try using a tried and true recipe that doesn't rely on the starter to provide all of the hydration.

Don't worry so much about gluten production and how sugar or fat will affect it. Small amounts of these enrichments, like those in the recipe you linked to, will soften the interior of the bread and help the crust brown nicely. Just look at brioche if you want an example of how far you can push enrichment and still get a well structured bread.