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Begin by cutting the beef chuck into 4-ounce pieces. Sprinkle lightly with salt and pepper. Heat the oil in a large heavy-bottomed pot over medium-high heat until hot. Brown the beef on all sides, and then remove. Turn down the heat to medium and add the tomato paste, celery and carrots and sprinkle with salt and pepper. Cook until the vegetables are sweating, about 5 minutes. Add the beef back into the pot, cover and transfer to the oven for 2 hours.
Transfer the beef to a baking sheet and let cool for 10 minutes. Shred the meat using a fork and tongs, and then return to the pot. Add the onions and peppers and place over medium heat. Cook until heated through, stirring occasionally.
Evenly divide the shredded beef mixture among the toasted brioche bun bottoms. Top each with a slice of cheese and a bun top. Place on a platter and enjoy.