Begin by cutting the cauliflower into chunks that will fit into your food processor. Pulse the cauliflower in your food processor until you have a rice-like texture. When you're finished you should have about 4 cups of riced cauliflower. You can save any extras for other dishes if you have extra.
Next heat a teaspoon of the vegetable oil in a large skillet over medium-high heat. When the oil is hot, add in the eggs and scramble. Then remove the eggs and set aside. Add in the remaining vegetable oil, garlic, and ginger and saute for a minute before adding in the peas, carrots, scallions, and cauliflower. Continue to stir-fry until the veggies are tender. This should take about 5 minutes.
In a separate bowl, whisk the sesame oil and soy sauce together. Then add in the soy sauce mixture and chicken to the fried rice and cook, stirring frequently, for 2-3 more minutes. Add in the eggs and give another stir to incorporate everything. Then remove from heat and serve with additional soy sauce if desired.