Pat the diced chicken breast dry and toss in a bowl with arrowroot powder, sea salt, and black pepper until evenly coated.
Heat the toasted sesame oil in a large non-stick skillet or wok over medium-high heat until shimmering.
Add the chicken to the skillet in a single layer and sear without moving for 3 minutes to develop a golden crust, then flip and cook for another 2-3 minutes until fully cooked.
Push the chicken to the outer edges of the pan and add the diced carrots and frozen peas to the center, sautéing for 2 minutes.
Stir in the minced garlic and fresh ginger, cooking for 30 seconds until the aromatics are fragrant.
Add the chilled cooked brown rice to the pan, breaking up any clumps with a spatula, and toss with the chicken and vegetables.
Create a small well in the center of the mixture and crack the egg into it, scrambling quickly until the egg is just set.
Drizzle the tamari over the entire dish and stir-fry for an additional 2 minutes, allowing the rice to slightly crisp against the hot pan.
Remove from heat and garnish with toasted sesame seeds and thinly sliced green onions before serving.