Prepare the cauliflower rice- Wash and thoroughly dry the cauliflower. Remove all the greens. If using a food processor, chop your cauliflower into small florets. Transfer to a large food processor, filling only 1/2-2/3 of the way full (you may need to process your cauliflower in 2 batches). Pulse the cauliflower in 1-second pulses until cauliflower is approximately the size of rice or couscous. If using a box grater, chop the cauliflower into large chunks and use the medium to large-sized holes to grate the cauliflower into "rice".
Cook the eggs - Heat a large skillet over medium-high heat. Add the olive oil and pour in the eggs. Cook the eggs, using a spatula to scramble them as they cook. Once cooked, remove from the skillet to a clean plate and set aside.
Sauté the vegetables - Return skillet to medium heat. Add the remaining tablespoon of olive oil followed by the carrots, red bell pepper, and peas. Cook the vegetables, stirring often, for 4-5 minutes.
Mix in the aromatics - Add the butter, garlic, fresh ginger, and white parts of the green onions. Cook for 1-2 minutes, stirring continuously.
Add the cauliflower rice - Stir in the cauliflower rice and mix well to combine with the vegetables. Continue to cook for 1-2 minutes, stirring often, before adding the soy sauce and sesame oil. Increase heat to high and cook the cauliflower fried rice until most of the liquid from the vegetables has evaporated, stirring as needed.
Return the eggs - Return the eggs to the skillet and mix well to combine. Drizzle with a couple of teaspoons of sesame oil, mixing well.
Season to taste - Season with additional soy sauce or salt and pepper, to taste.
Serve - Serve garnished with the green parts of the green onions, sesame seeds, lime wedges, and hot sauce, if desired.