Add your warm water (roughly 120 degrees or less) to your large mixing bowl (or the bowl of a stand mixer). Mix in active dry yeast and sugar. Allow to rest for roughly 5 minutes while the yeast blooms.
Next, add in warm milk, sourdough discard, softened butter, and 1 egg + 1 egg yolk (preferably room temperature). Mix until combined.
Add in the flour and salt. Mix together by hand or in a stand mixer on low speed until a smooth ball is formed.
Knead for 5-10 minutes or until the dough passes the "windowpane test" (where you can stretch the dough without it breaking but can see light through it). Don't worry about over-kneading.
Allow to rest in a greased bowl for roughly an hour or until it has doubled in size.
While the dough is rising, make your garlic and herb butter. In a bowl, mix together the butter, mozzarella, parmesan, salt, dried garlic, dried parsley, and dried thyme. Set aside.
Once the dough has fully proofed, place it on a lightly floured surface and separate into 12 equal-sized balls. Each ball will be roughly 70 grams in weight if you want to be technical.
Flatten each ball. Don't worry about making them particularly uniform in shape. Place roughly 1 tablespoon of your butter mixture on each flattened dough ball.
Lightly spread to cover all of the dough. If you have any leftover mixture, you can melt it and spread it over the loaf prior to baking for extra flavor.
Fold dough over the mixture like a taco and lightly stretch to make a rectangle (you'll want it to be about as wide as your loaf pan)
Place each of the 12 pieces of dough in a buttered loaf pan one after the other. Preheat your oven to 375 degrees Fahrenheit and allow the dough to rest while it preheats. Don't worry about how much it rises.
Bake the sourdough pull-apart bread for 40-45 minutes or until it is golden brown and fully baked through.
Top with melted butter and enjoy warm!