Salt brine the chuck roast - Place the chuck roast on a wire cooling rack set on top of a sheet tray. Season all sides of the roast with kosher salt. Refrigerate uncovered for at least 8 hours or overnight, up to 24 hours.
Prepare the braising liquid - To a half-size deep foil pan or a Dutch oven, add the can of pickled jalapeños with the juice, tomato sauce, and minced garlic. Add the sliced onions and bell peppers in an even layer. Evenly sprinkle on the Fiesta That and Spice That seasonings. Place the stick of butter on top. Set to the side.
Preheat the grill or smoker to 250℉ with your choice of wood or pellets. *For oven braising, preheat the oven to 325℉.
Sear the beef roast - Heat your griddle to about 475°F. Or for stovetop searing, preheat a large skillet over medium heat. Add about 1 tablespoon of high heat cooking oil. Once hot, place the chuck roast on the cooking surface, pressing down slightly to ensure good contact. Sear untouched for about 4-5 minutes on the first side, or until a nice dark crust develops. Flip to the second side, and sear an additional 3-4 minutes.
Transfer the seared beef to the prepared braising liquid, laying the roast on top of the vegetables. Pour in about 1 cup of beef stock, or enough to submerge the roast about halfway.
Cook the beef roast - Place the braising pan with the beef on the grill grates, and smoke uncovered at 250℉ for 2 hours. Then cover the pan with aluminum foil, bump up the grill temp to 325℉, and braise covered until the internal temperature of the beef reaches about 205-207℉. *My total cooking time at 325℉ was about 3½ hours, but times will vary. Cook the roast to the proper internal temp and fork tenderness, rather than to a specific time. **See notes for oven braising.
Transfer the beef to a cutting board to cool about 4-5 minutes before shredding. While the roast is cooling, skim off the excess fat from the top of the braising liquid. Then strain to separate out the peppers and onions, reserving the flavorful broth and the cooked vegetables separately.
Use a fork to shred the Mexican beef into small bite-sized pieces. As you’re shredding, remove the larger pieces of fat and discard.
Heat griddle or skillet on low to medium-low heat.
Place one tortilla down, sprinkle generously with cheese, add some of the shredded beef, top with some of the reserved vegetables, and (if desired) drizzle on a little of the braising sauce. Add a final sprinkle of cheese, then fold the tortilla over to form the quesadilla. Cook on the first side about 3-4 minutes or until golden brown. Then flip to the second side and continue to cook until evenly golden and cheese is melted.
Transfer to a cutting board to rest for 2 minutes before slicing into triangles. Serve with reserved braising liquid for dipping, or sour cream, guacamole, or your favorite toppings.