I don't think you're doing anything wrong, I think the dough is just more slack than you're used to. As @Jay noted, it can take some practice to work with a wet dough. But once you do, you'll be rewarded with a much more open crumb and a better final product. In my experience, I've found wetter dough and higher oven temps = better artisan bread (in general).
The recipe appears to assume the reader is familiar with the process, but does offer some hints. She talks about scraping the dough out onto the work surface
, then stretching-and-folding
.
The recipe isn't as wet as the ciabatta I'm making below, but the process will be similar, so I hope this is helpful. I start by putting a bed of flour down, then scraping the blob of wet dough out onto it.
Then comes the stretch-and-fold part, which is just as it sounds. Using a wet pastry scraper and/or wet hands, just get under one edge, lift and pull it away, then plop it back on top of the main dough blob. Then do the same with the other side. Cover with plastic wrap and walk away. There's no process of kneading like you're used to. This photo is after a few stretch-and-folds at 20-minute intervals (I think!), and you can see the dough has started to smooth out and become cohesive.
By the time you're ready to shape, the dough should be a lot more cohesive and easier to deal with. I folded mine into little slippers and put them on a couche to rise.
Add 500 degrees and a baking stone, and I'm rewarded with an open and gelatinized crumb, and a nice crisp crust.
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.
Best Answer
If you are making sourdough you are working with natural yeasts in an acidic dough, so I wouldn't think adding parmesan and garlic is going to be a problem as long as you leave extra time for proofing. Parmesan is salty, and salt inhibits yeast. You are probably adding salt already so reducing the amount to compensate would make sense.
As for adding garlic in my experience it doesn't work that well just chopped up and in the bread whether you add it at the beginning or knead it in later. The flavor doesn't get out and you get chunks of intense garlic rather than a nice hum in every bite, although maybe that's the effect you want. I have a garlic and rosemary infused olive oil I use rather than adding garlic directly, another option is to put chopped garlic in olive oil (or butter, or any other oil) on very low heat for 10-15 minutes, this will infuse the oil with the garlic flavor which you add to the break. I haven't had any issues with this inhibiting yeast.
I would add these ingredients in at first mix rather than kneading it in later as kneading after first proof as you will knock out most of your air and damage the structure. You may need a longer proof but the structure will be better. Alternatively if your dough is stretchy enough (like a pizza dough or focaccia) you can stretch it flat after first rise, put ingredients in as a layer, then fold it over, then do your final rise and bake. That only works when you have a stretchy dough though.