If making the pickled things, in a medium bowl, add the broccoli stem and carrot, vinegar, water, and salt. Set aside.
In a large non-stick skillet over medium-high heat, add one tablespoon oil. Once the oil is hot and shimmering, add the white parts of the scallions, carrot, and ¼ teaspoon salt. Cook for 2-3 minutes, stirring occasionally, until softened and a little browned in spots. Add the cabbage and cook for 2 more minutes, or until softened, stirring often. Transfer to a large bowl or container (or just move the veg to the side and add the broccoli, see note).
Return the skillet to medium-high heat and add a drizzle of olive oil (about 1 teaspoon), broccoli and ¼ teaspoon salt and cook for 1 more minute, stirring occasionally, until broccoli is tender and browned in spots. Transfer to the bowl.
Return the skillet to medium-high heat, add 1 tablespoon oil, and swirl to coat the pan. Add the eggs and sprinkle with remaining ¼ teaspoon salt and let cook for 30 seconds, undisturbed, then scramble them up by pulling the sides in and folding the eggs. Once the eggs have formed curds and are mostly cooked through, break them up into smaller curds/chunks and transfer to the bowl.
Return the skillet to medium-high heat and add remaining 1 tablespoon oil and cooked rice. Spread into an even layer and cook, undisturbed, for 2 minutes, to lightly toast the rice. If the rice is really dry, you can add a splash of water to hydrate.
Stir the veggies and eggs back in along with the ginger, stirring to combine everything. Remove from heat and stir in the soy sauce or tamari, rice vinegar, and half of the green parts of the scallions.
Serve topped with the remaining scallions, peanuts or cashews, pickled veggies (if using), and a squeeze of lime. Also delicious topped with sriracha or chile crisp.