To make the barbecue sauce, in a large saucepan, whisk together the ketchup, agave syrup, brown mustard, the 1/2 cup vinegar, Worcestershire sauce, garlic salt, ground mustard, onion powder, chili powder, bay leaf, Frank's RedHot Sauce, horseradish, the 1/2 cup beer, onion, liquid smoke and cayenne until well combined. Set over medium heat and bring to a slow boil, then reduce the heat to low and simmer, uncovered, for 1 hour. Add more vinegar or beer, to taste. 

Let the sauce cool to room temperature, then cover and refrigerate overnight.

To make the brioche buns, in a glass measuring cup, combine the warm water, milk, yeast and granulated sugar. Let stand until foamy, about 5 minutes. Meanwhile, in a small bowl, beat 2 of the eggs.

In a large bowl, whisk together the 6 cups bread flour, the all-purpose flour and salt. Add the butter and rub it between your fingers into the flour, making crumbs. Using a dough scraper, stir in the yeast mixture and the beaten eggs until a dough forms. Scrape the dough onto a clean, well-floured counter and knead, scooping the dough up, slapping it on the counter and turning it, until smooth and elastic, 8 to 10 minutes. The dough will somewhat sticky, so it can be a bit messy, but keep in mind that the more flour you knead in, the tougher the buns will be. Try to leave them tackier than you would a round loaf.

Shape the dough into a ball and return it to the bowl. Cover the bowl with plastic wrap and let rise in a warm place until doubled in bulk, about 1 hour.

Preheat the oven to 375°F. Line a baking sheet with parchment paper.

Roll out half of the dough about 1/2 inch thick. Using a 1 1/2-inch cookie cutter, cut into rounds. Roll each dough round into a ball and flatten slightly with your hand. Place on the prepared baking sheet, spacing them about 2 inches apart. Repeat with the other half of the dough.

In a small bowl, beat the remaining 2 eggs. Brush the tops of the buns with the beaten eggs. Bake until the buns are golden brown, 15 to 18 minutes. 

To make the pulled pork, let the pork shoulder stand at room temperature for 1 hour.

In a bowl, whisk together 1/2 cup of the apple juice, 1/4 cup of the water, 1/4 cup of the granulated sugar, 2 tablespoons of the kosher salt and the Worcestershire sauce to make the injection marinade. Inject the pork evenly with a marinade injector.

In a bowl, stir together the remaining 1/4 cup granulated sugar, the dark brown sugar, paprika, garlic salt, the remaining 2 1/2 tablespoons kosher salt, the chili powder, oregano, cayenne, cumin and black pepper to make the rub. Generously rub the pork with the mixture, patting it so the seasonings adhere.  

Preheat a smoker/grill to 225°F. 

Place the pork shoulder, fat side up, directly on the grill and cook for 2 1/2 hours. Increase the grill temperature to 250°F. Put the pork in a disposable foil pan and return it to the grill. In a spray bottle, combine the remaining 1/2 cup apple juice and 1/4 cup water and spray the pork thoroughly every hour or so. Cook until an instant-read thermometer inserted into the center of the pork registers 190°F, 5 to 7 hours more. If the pork begins to brown too much, covering loosely with aluminum foil. 

Transfer the pork to a cutting board and let rest for 30 minutes. Wearing rubber gloves, pull the meat apart. Add some of the juices from the aluminum pan to the shredded pork and add barbecue sauce as desired. Serve the pork on the brioche buns.