Whisk together tamari, vinegar, and honey in a small bowl.
Warm 1 tablespoon avocado oil in a large skillet over medium-high heat. Add mushrooms, sprinkle lightly with salt, and cook, stirring occasionally, until they release their water and become tender, 3 to 4 minutes. Add 1 tablespoon avocado oil and carrots, celery, and bell pepper; sauté until vegetables are tender, about 2 minutes.
Stir in garlic, ginger, and scallions; sauté 30 seconds. Add bok choy and cook, stirring often, until bok choy wilts, 1 to 2 minutes. Add remaining 1 tablespoon avocado oil and cauliflower rice. Cook, stirring occasionally, until rice begins to get tender, about 1 minute. Stir tamari mixture into skillet; cook, stirring occasionally and scraping up browned bits from bottom of skillet, until liquid cooks off and rice is tender, about 2 minutes.
Remove from heat and drizzle sesame oil on top. Add a few shakes of sriracha, if desired, and sprinkle with sesame seeds. Divide fried rice among 4 plates or bowls and serve.