Cook the chicken: Preheat the oven to 425°F and line a large baking sheet with parchment paper. Set aside. Transfer the chicken breasts to a large mixing bowl and drizzle with olive oil. Season with Italian seasoning, salt, and pepper. Place the chicken on the prepared baking sheet and roast for 20-22 minutes, or until the internal temperature reads 155-165°F with a digital thermometer. Remove from the oven and rest for 5-10 minutes before chopping or shredding. See notes (1). Boil a large pot of water: Bring a large pot of water to a boil over high heat. We will need this in step 4.
Cook the rice (to al dente). Thoroughly rinse the rice in several changes of clean water. Add the rice to a medium saucepan with 1-2 teaspoons of salt and enough water that the rice is covered by at least 1 to 2 inches of water. Bring to a boil, then immediately cover and reduce the heat to low. Check the progress of the rice every 2-3 minutes by pressing a few grains between your fingers. You know it's ready when it squishes apart. Drain any remaining liquid from the rice and rinse with cool water (to stop the rice from cooking further), then drain again. Transfer back to the pot and set aside. See notes (2)
Blanch the broccoli. Add the broccoli florets to the pot of boiling water. Blanch (boil) for 2-3 minutes, careful not to overcook. Drain, then immediately transfer to a large bowl filled with ice water (ice bath) to stop cooking. Pat dry with paper towels and set aside. See notes (3)
Preheat oven to 350°F and spray a large baking dish with non-stick cooking spray. Set aside.
Melt two tablespoons of butter in a large pan (with high sides) over medium heat. Add the onions and cook until soft and translucent, 6-7 minutes. Stir in the minced garlic, salt, and black pepper. Cook for an additional 30-60 seconds, stirring continuously. See notes (4)
Make the sauce. Melt two more tablespoons of butter in the pan and then sprinkle with flour (this will make the roux). Whisk continuously to combine. Still whisking, add the milk slowly. Continue to whisk until thickened and bubbly (do not boil). Remove from heat and add the Dijon mustard, softened cream cheese, and one cup of shredded cheddar cheese. Stir until combined and the cheese has melted. See notes (5&6) Assemble: Transfer the prepared chicken, broccoli, and rice to the pan and mix well to combine. Season with additional salt and pepper to taste, then transfer to your prepared baking dish.
Sprinkle the remaining shredded cheese and crushed buttery crackers over the top (if using).
Bake for 30-35 minutes, or until hot and bubbly around the edges and the cheese is lightly browned- if the cheese and cracker topping starts to burn, loosely tent the baking dish with foil.
Rest and serve: Remove from the oven and rest for 5-10 minutes before serving.