In a small bowl, whisk together the coconut aminos, honey, grated ginger, and minced garlic to create the base sauce.
In a separate small ramekin, stir the arrowroot powder and water together until smooth to create a slurry and set aside.
Heat the olive oil in a large skillet or wok over medium-high heat.
Season the chicken breast pieces with sea salt and black pepper, then add them to the skillet and sauté until golden and cooked through, about 5-7 minutes.
Remove the chicken from the skillet and set aside; add the sesame oil and broccoli florets to the same pan.
Add a splash of water to the broccoli, cover the pan with a lid, and steam for 2 minutes until the broccoli is bright green and tender-crisp.
Return the chicken to the skillet and pour the sauce mixture over the ingredients, tossing to coat everything thoroughly.
Stir in the arrowroot slurry and simmer for 1 minute until the sauce thickens into a glossy glaze.
Serve the teriyaki chicken and broccoli immediately over the warm cooked brown rice.