Preheat the oven to 200°C (400°F) and line a roasting tray with foil or parchment paper.
Place the peppers, onion, and garlic on the tray, brush lightly with avocado oil, and roast for 20 to 25 minutes, until softened and lightly charred.
Transfer the peppers to a bowl, cover, and let steam for 10 minutes. Peel off and discard the skins.
Add the peeled peppers, onion, garlic, salt, black pepper, oregano, and paprika to a blender. Blend until smooth, then strain if you want a silky sauce. Set aside.
Combine all the dry rub ingredients in a small bowl and season the chicken evenly on both sides.
Heat the avocado oil in a sauté pan over medium-high heat. Cook the chicken until golden and cooked through, about 2 minutes per side if thinly cut, or about 4 minutes per side for whole breasts. Remove and set aside.
Reduce the heat to medium and add the red pepper sauce to the pan, scraping up any browned bits. Bring to a gentle simmer.
Stir in the Parmesan and heavy cream and cook until the sauce thickens slightly.
Return the chicken to the pan and spoon the sauce over the top. Warm through gently.
Finish with chopped parsley and serve immediately.