Using a large skillet, melt 1 tablespoon of butter over medium heat. Once hot, add the beaten eggs, stirring only occasionally until soft scrambled. Remove from the pan and set aside.
Melt 2 tablespoons of butter in the skillet and increase the heat to medium-high. Add the diced carrots, onion, and peas and stir them into the butter. Cook the veggies, stirring occasionally, until slightly brown, about 3-5 minutes. Add the garlic and allow to cook for 1 minute.
Push the sauteed veggies to one side of the pan. Add the remaining tablespoon of butter to the middle of the pan, and then add the cauliflower rice. Give the cauliflower a quick stir to distribute the butter. Then allow it to brown slightly by not stirring for a couple of minutes. Once the bottom of the cauliflower starts to brown, stir all the veggies together and cook them for another 2-3 minutes.
Stir in the green onions, soy sauce, sesame oil, sesame seeds, and crambled egg. Remove from the heat. Garnish with additional sesame seeds and green onion, if desired. Serve warm.