Prepare the Components: On a cutting board, cover the chicken breasts with parchment paper and gently pound them to a uniform ¾-inch thickness. In a small bowl, combine the garlic powder, oregano, smoked paprika, salt, and pepper. Season all sides of the chicken with this mixture. Chop the broccoli into bite-sized florets.
Steam the Broccoli: Pour ½ cup of the chicken broth into the braiser pan and bring to a simmer over medium heat. Add the broccoli florets, cover with the lid, and allow to steam for 5–6 minutes, or until bright green and tender-crisp. Transfer the steamed broccoli to a bowl and set aside, discarding any remaining liquid.
Sear the Chicken: Wipe the pan dry and return it to the stovetop. Increase the heat to medium-high and add the avocado oil. Once shimmering, carefully place the seasoned chicken in the pan. Sear for 4–5 minutes on each side until a deep golden-brown crust forms. The chicken does not need to be fully cooked at this stage.
Build the Sauce Foundation: Reduce the heat to medium-low. Move the chicken to one side of the pan and pour the remaining ½ cup of broth into the empty space, using a whisk to scrape up any flavorful browned bits from the pan's surface. Add the softened cream cheese chunks and whisk them directly into the broth until a smooth, uniform sauce begins to form.
Melt the Cheeses: Sprinkle the shredded white cheddar and grated Pecorino-Romano into the sauce. Continue to whisk gently until all the cheese has melted and the sauce is creamy and consistent.
Unify and Serve: Return the steamed broccoli to the pan, arranging it around the chicken. Spoon some of the sauce over everything. Cover the pan and let the dish gently heat through for another 2–3 minutes, allowing the flavors to meld and ensuring the chicken reaches an internal temperature of 165°F (74°C). Serve immediately, spooning extra cheese sauce over each portion.
