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Sprinkle chuck roast and short ribs all over with 2 tablespoons salt. Combine roast, ribs, and adobo in a large nonreactive bowl; toss to coat. Cover and chill at least 4 hours or up to 24 hours.
Preheat oven to 300°F. Transfer adobo mixture to a large (9 1/2-quart) Dutch oven; add 8 cups water. Bring to a simmer, uncovered, over medium, stirring occasionally. Cover with lid, and place in preheated oven. Bake until meat is fork- tender, about 4 hours.
Remove chuck roast and short ribs from braising broth, and transfer to a large bowl; cover with aluminum foil to keep warm. Return broth in Dutch oven to heat over medium, and cook, uncovered, skimming off fat as needed, until reduced to about 8 cups, 15 to 20 minutes. Season broth with salt to taste. Shred meat; discard bones. Toss meat with 1 1/2 cups of the broth.
Stir together onion, cilantro, and remaining 1/4 teaspoon salt in a small bowl; set aside.
Heat a large nonstick electric griddle to 400°F or a large (12-inch) cast-iron skillet over medium-high. Using a paper towel dipped in canola oil, lightly grease griddle. If using fresh tortillas, stack 2 tortillas, and use tongs to dip them together into adobo broth. (If using packaged tortillas, dip 1 tortilla per taco.) Place stacked tortillas on griddle; top with 1/4 cup meat. Repeat with as many tortilla stacks as will comfortably fit on griddle. Cook until bottom tortilla is lightly browned and crispy, 1 to 2 minutes. Fold tacos in half, gently pressing with a spatula. Transfer to a serving plate. Repeat process with oil, adobo broth, remaining tortillas, and remaining meat.
Serve tacos hot with onion-cilantro mixture, lime wedges, and remaining adobo broth for dipping or sipping.