Heat 1 tablespoon of the coconut oil in the cleaned skillet over high heat. When hot, add the chicken in an even layer. Cook until golden brown on the first side, about 2 minutes, then stir fry until the chicken is cooked through, about 4 minutes more. Transfer to a bowl and set aside.
Add the remaining vegetable or coconut oil and stir fry the peppers and white scallions until lightly charred, 2 minutes. Stir in the garlic and cook until fragrant, 1 minute. Add the chicken back to the pan, along with the tamari, sesame oil, honey, and sesame seeds. Simmer until the sauce has thickened, 2 minutes. Stir in the broccoli and green scallions. Serve over brown rice or quinoa.
If you find that the chicken sticks to the skillet, you can added an additional 1/2 cup of water or chicken stock with the sauce to help deglaze the pan. Usually, the peppers are enough to release those brown bits.