Preheat oven to 350° F. Heat a large skillet with a lid over medium heat. Add the broccoli florets and a 1/2 cup of water to the skillet. Cover with a lid and let the broccoli steam for 3-5 minutes or until bright green and crisp tender. Remove the broccoli from the skillet and onto a plate. Drain any excess water from the skillet.

Add the olive oil to the skillet and heat it over medium-high heat. Add the onion, mushrooms, salt and pepper to the skillet. Sauté for 3-4 minutes and then add in the garlic, smoked paprika, thyme leaves. Sauté another minutes and then sprinkle the flour over everything.

Stir until you can no longer see streaks of flour. Slowly pour the milk in stirring the entire time. Continue to stir until the sauce starts to thicken. Add the broccoli, rice, and half of the cheese into the skillet. Stir everything together.

Remove the skillet from the heat and stir in the Greek yogurt until thoroughly combined. Top the casserole with the remaining cheese and cover with the lid. Bake for 15-20 minutes and then remove the lid and bake another 5 minutes to crisp up the cheese.