Preheat oven to 350°F. Lightly spray a 9×13-inch baking dish with nonstick cooking spray and set aside.
In a small bowl, mix together paprika, salt, pepper, and garlic powder. Season both sides of the chicken breasts evenly, pressing the seasoning into the meat.
Place the flour in a shallow dish and dredge each chicken breast, coating all sides. Shake off any excess flour.
In a large skillet over medium heat, melt 4 tablespoons of the butter. Add the chicken (working in batches if needed) and cook for about 4 minutes per side, until golden brown. The chicken does not need to be fully cooked through at this stage. Transfer to the prepared baking dish.
In the same skillet, melt the remaining 2 tablespoons butter. Add the diced onion and cook for 5 to 8 minutes, until softened.
Add the broccoli and cook for 3 to 4 minutes, stirring occasionally, until it just begins to soften.
Stir in the cream of chicken soup, heavy cream, and 1 cup of the shredded cheddar cheese. Stir until smooth and creamy.
Pour the broccoli cheese sauce evenly over the chicken in the baking dish.
Bake uncovered for 40 minutes.
Remove from the oven, sprinkle the remaining 1 cup of cheddar cheese over the top, and return to the oven. Bake for an additional 5 to 10 minutes, or until the cheese is melted and bubbly and the chicken reaches an internal temperature of 165°F.
Let rest for 5 to 10 minutes before serving so the sauce can settle. Spoon extra sauce over each serving.