Shred the cauliflower using the large holes on a box grater or pulse in the food processor until the cauliflower resembles large grains of rice. (You can now purchase already “riced” cauliflower in most supermarkets.) Set aside.
In a small bowl, lightly beat the eggs with ½ teaspoon of tamari and set aside.
Heat 1 tablespoon vegetable oil in a large skillet or wok over medium heat. Add the carrots and cook, stirring, until crisp tender, about 5 minutes. Add the garlic and cook, stirring for another 30 seconds. Add the cauliflower, chicken, edamame, green onions, sesame oil, ginger, and remaining tamari and cook, stirring, until everything is heated through, about 5-6 minutes.
Make a well in the center of the mixture by pushing everything to the sides of the pan. Add the remaining 1 tablespoon of vegetable oil, let it heat for about 15 seconds, then add the eggs. Scramble the eggs, then mix them into the rest of the mixture.