Prepare the ingredients by chopping the cooked beef brisket into small bite-sized pieces, dicing the yellow onion, mincing the garlic, and slicing the green onions, keeping the white and green parts separate.
Break up any clumps in the chilled cooked long-grain white rice with clean hands or a fork so the grains are separated and ready for stir-frying.
In a small bowl, whisk together the low-sodium soy sauce, oyster sauce, and dark soy sauce (if using), then set this sauce mixture aside near the stove.
Heat 1 tablespoon of the neutral cooking oil in a large wok or wide skillet over medium-high heat until the oil is shimmering.
Pour the lightly beaten eggs into the hot pan and quickly scramble them with a spatula, cooking just until softly set, then transfer the scrambled eggs to a plate and set aside.
Add the remaining 1 tablespoon of neutral cooking oil to the same pan, then add the diced yellow onion and the white parts of the green onions, cooking and stirring for 2 to 3 minutes until they begin to soften and turn translucent.
Stir in the minced garlic and cook for about 30 seconds, just until fragrant, being careful not to let it brown.
Add the chopped cooked beef brisket to the pan and stir-fry for 2 to 3 minutes, allowing the meat to warm through and develop a bit of caramelization on the edges.
Add the thawed peas and carrots mix to the pan and cook for another 1 to 2 minutes, stirring, until the vegetables are heated through.
Increase the heat to high, then add the broken-up chilled rice to the pan, spreading it out and using a spatula to press and toss so the grains are coated with the aromatics and oil.
Pour the prepared sauce mixture evenly over the rice and brisket, then stir-fry for 2 to 3 minutes, tossing continuously so the rice absorbs the sauces and takes on an even color.
Return the scrambled eggs to the pan, breaking them into smaller pieces with the spatula, and gently fold them through the rice mixture.
Drizzle the toasted sesame oil over the fried rice, then season with freshly ground black pepper and a small pinch of kosher salt, tasting and adjusting the seasoning as needed.
Stir in the sliced green parts of the green onions, reserving a small handful for garnish if desired, and toss everything together for another 30 seconds over the heat.
Remove the pan from the heat and serve the brisket fried rice hot, garnished with the remaining green onion tops if you like, and enjoy immediately.