Rice the cauliflower: cut the head into florets and pulse in a food processor in 2 batches until the pieces are the size of rice grains, about 10 pulses per batch (or grate the florets on the large holes of a box grater). You should have about 6 cups; set aside.
Heat 1 tbsp of the oil in a 12-inch nonstick skillet over medium heat. Add the eggs and scramble until just set, about 1 minute, breaking them into small pieces. Transfer to a plate.
Add the remaining 1 tbsp oil and raise the heat to medium-high. Add the carrots and bell pepper and cook, stirring, for 3 minutes. Stir in the scallion whites, garlic, and ginger and cook for 30 seconds, until fragrant.
Add the riced cauliflower and the 1/2 tsp salt. Cook for 8-10 minutes, stirring often and spreading the mixture out in the pan, until the cauliflower is tender and its moisture has cooked off.
Stir in the soy sauce and sesame oil and cook for 1-2 minutes more. Return the eggs, add the scallion greens, and toss to combine. Taste and add more soy sauce if needed.
Divide among 4 bowls and serve hot.