Preheat oven to 375°F (190°C).
Lightly grease a 9×13-inch casserole dish with a small amount of the 1 Tbsp olive oil (reserve the rest for cooking).
In a large skillet over medium heat, heat the remaining olive oil and sauté the minced garlic for about 1 minute, until fragrant (do not let it brown).
Add the 1 3/4 cups low-sodium chicken broth, 1/2 cup heavy cream, juice of 1 lemon, 1 tsp fresh thyme (or rosemary), 1/2 tsp red pepper flakes, and salt and pepper to taste. Bring the mixture to a gentle simmer.
Stir in 1/2 cup of the grated Parmesan cheese until combined, then remove the skillet from the heat.
In the prepared casserole dish, combine the 3/4 cup uncooked long-grain white rice, 2 cups broccoli florets, and the 1 lb chicken breasts cut into 1 1/2-inch pieces. Season the chicken with additional salt and pepper if desired, and spread everything evenly.
Pour the hot sauce from the skillet evenly over the rice, broccoli, and chicken, making sure the rice is submerged in the liquid and ingredients are coated.
Sprinkle 1/3 cup breadcrumbs over the top, then sprinkle the remaining 1/2 cup grated Parmesan cheese evenly over the breadcrumbs.
Tightly cover the casserole with foil and bake in the preheated oven for 35 minutes.
Remove the foil and bake an additional 10 minutes, uncovered, until the chicken is cooked through (no pink inside or reaches 165°F) and the rice is tender and has absorbed the liquid.
Remove the casserole from the oven and let it rest for 5 minutes before serving.