Combine 1 teaspoon sesame oil with eggs in a bowl then whisk to combine and set aside. Chop cauliflower head into big chunks then send through a food processor fitted with the grating attachment. Alternatively you could grate the cauliflower on a box grater. Measure out 4 cups grated cauliflower then save the rest for another use.
Heat remaining 1 Tablespoon sesame oil in a wok or large nonstick skillet over high heat. Add cauliflower, green onions, and frozen mixed vegetables then stir fry until cauliflower is just beginning to turn tender, 3-4 minutes. Add garlic, season with salt and pepper, and then continue to stir fry until garlic is fragrant, 30 seconds or so.
Push mixture to the sides of the wok to create an opening in the center then add the eggs and scramble. Toss mixture to combine then drizzle in gluten-free Tamari and mix again to combine. Serve hot.