Pat the beef brisket cubes dry with paper towels and season them lightly with a pinch of the kosher salt and black pepper.
Heat 1 tablespoon of olive oil in a large heavy-bottomed pot or Dutch oven over medium-high heat.
Add half of the brisket cubes in a single layer and sear until browned on at least two sides, about 4–5 minutes per batch. Transfer the browned brisket to a plate.
Add the remaining 1 tablespoon of olive oil to the pot and repeat the searing process with the remaining brisket cubes. Transfer all browned brisket to the plate and set aside.
Reduce the heat to medium and add the diced yellow onion to the same pot. Cook, stirring occasionally, until the onion softens and becomes translucent, about 4–5 minutes.
Stir in the diced red bell pepper and cook for another 3–4 minutes until slightly tender.
Add the minced garlic and cook for 30 seconds to 1 minute, stirring constantly, until fragrant but not browned.
Sprinkle in the chili powder, ground cumin, smoked paprika, dried oregano, remaining kosher salt, black pepper, and crushed red pepper flakes if using. Stir well to coat the vegetables in the spices and toast them for about 1 minute.
Add the tomato paste to the pot and cook, stirring, for 1–2 minutes to deepen its flavor and slightly caramelize it.
Pour in the crushed tomatoes and beef broth, stirring to combine and scraping up any browned bits from the bottom of the pot.
Return the seared brisket cubes and any accumulated juices from the plate back into the pot, stirring them into the tomato and broth mixture.
Stir in the kidney beans, black beans, and brown sugar until everything is evenly distributed.
Bring the chili to a gentle boil over medium-high heat, then reduce the heat to low to maintain a steady simmer.
Cover the pot with a lid, leaving it slightly ajar, and simmer the brisket chili for about 2.5 to 3 hours, stirring occasionally, until the brisket is very tender and the flavors are well developed.
Once the brisket is fork-tender, taste the chili and adjust seasoning with additional salt or black pepper if desired.
If you prefer a slightly thicker chili, remove the lid and simmer uncovered for an additional 10–15 minutes, stirring occasionally, until it reaches your desired consistency.
Turn off the heat and let the brisket chili rest for about 10 minutes to settle before serving.