Prepare the barbacoa: Process chiles, garlic, broth, vinegar, brown sugar, cumin, oregano, chipotle chiles, 1 teaspoon of the black pepper, and 1 1/2 teaspoons of the kosher salt in a blender until smooth. Set aside.
Heat olive oil in a large skillet over medium-high. Sprinkle chuck roast with the remaining 1 1/2 teaspoons salt and 1 teaspoon black pepper. Add roast to the skillet; cook until brown on both sides, about 5 to 6 minutes per side. Transfer to a 5-quart slow cooker; add chile mixture. Cover and cook on LOW until the meat is very tender, about 8 hours.
Remove roast from slow cooker, reserving cooking liquid in slow cooker. Skim any excess fat from cooking liquid. Using 2 forks, shred meat and return it to the liquid in the slow cooker. Keep warm until ready to serve.
Prepare the puffy tacos: Shape masa dough into 8 (1 3/4-inch) balls. Line bottom of a tortilla press with parchment paper. Place 1 masa ball on the parchment, and top with another piece of parchment; close tortilla press to form a 5 1/2-inch tortilla. (Or press ball between sheets of wax paper against a countertop using a flat plate or skillet.) Repeat with remaining masa dough balls. Gently stack tortillas between layers of parchment paper, and chill until ready to fry.
Pour vegetable oil into a Dutch oven, and heat to 375°F. Gently lower 1 uncooked tortilla into hot oil. Using a metal or wooden spatula, lightly press center of tortilla to create a "U" shape. Cook until golden brown, about 30 seconds. Carefully remove from oil, and sprinkle with desired amount of salt. Drain upside down on a paper towel-lined wire rack. Repeat procedure with remaining tortillas, and keep warm in a 200°F oven.
Prepare the avocado cream: Process all Avocado Cream ingredients in a blender until smooth, about 45 seconds. Chill until ready to use.
Divide beef among tortillas (about 4 ounces each); top evenly with onion, cilantro, and avocado cream. Serve immediately.