Pat the beef brisket cubes dry with paper towels to help them brown well. Season the brisket evenly with 1 teaspoon of kosher salt and 1/2 teaspoon of freshly ground black pepper.

Heat the olive oil in a large heavy-bottomed pot or Dutch oven over medium-high heat. When the oil is hot and shimmering, add the seasoned brisket cubes in a single layer, working in batches if needed to avoid overcrowding.

Brown the brisket cubes on all sides, turning occasionally, for about 8 to 10 minutes per batch, until they develop a deep, rich color. Transfer browned brisket to a plate and set aside while you continue with the remaining meat.

Reduce the heat to medium. In the same pot with the browned bits on the bottom, add the diced yellow onion. Cook, stirring often, for 4 to 5 minutes, until the onion softens and becomes translucent.

Stir in the minced garlic and cook for about 30 seconds to 1 minute, just until fragrant, being careful not to let it burn.

Add the sliced carrots and celery to the pot. Cook for 3 to 4 minutes, stirring occasionally, to begin softening the vegetables and to coat them with the flavorful cooking fat.

Return the browned brisket and any accumulated juices on the plate back into the pot with the vegetables, stirring gently to combine everything.

Pour in the diced tomatoes with their juices and the low-sodium beef broth, stirring to loosen any browned bits from the bottom of the pot for extra flavor.

Add the bay leaves, dried thyme, smoked paprika, dried oregano, and red pepper flakes if using. Stir well so the seasonings are evenly distributed throughout the soup.

Increase the heat to bring the mixture up to a gentle boil. Once it reaches a boil, immediately reduce the heat to low so the soup simmers steadily but not vigorously.

Cover the pot with a lid, leaving it slightly ajar, and let the brisket soup simmer on low heat for about 90 minutes, stirring occasionally, until the brisket begins to feel tender when pierced with a fork.

After about 90 minutes of simmering, add the cubed russet potato to the pot, stirring to submerge the pieces in the broth.

Continue to simmer the soup uncovered or partially covered for another 30 to 40 minutes, or until the potatoes are tender and the brisket is very soft and easy to break apart with a fork.

Once the brisket is fully tender, use a spoon to skim off any excess fat from the surface of the soup, if desired, to keep the broth balanced and not overly rich.

Stir in the chopped fresh parsley and lemon juice, then taste the soup. Add the additional 1/2 teaspoon of kosher salt and 1/4 teaspoon of black pepper, or more as needed, adjusting the seasoning to your preference.

Remove and discard the bay leaves. Ladle the brisket soup into bowls and serve hot, making sure each serving includes a good mix of brisket, vegetables, and broth.