Preheat your oven to 350°F and coat a 1.5 quart casserole dish with cooking spray.
Place a steamer basket in a large pot over simmering water. Add the cauliflower florets and stem for 12-15 minutes, until very tender.
While the cauliflower is cooking, heat a skillet over medium heat.
Add the ground beef, onion, garlic, 1 teaspoon salt and 1/4 teaspoon pepper to the skillet and cook, breaking up the meat, until it is browned. Drain any excess grease.
Add the mushrooms and green beans, and saute for about 5 minutes until the mushrooms are tender.
Stir in the beef broth tomato paste, and Worcestershire sauce. Bring to a simmer, and cook for 10-15 minutes or until thickened slightly. Season to taste with salt and pepper.
While the meat mixture is simmering, remove the cauliflower from the pot and allow some of the excess moisture to evaporate.
Transfer cauliflower to a food processor or blender. Add Greek yogurt, egg, ½ cup shredded cheddar cheese, and 1 teaspoon salt to the potatoes. Blend or puree until smooth, stopping to scrape down the sides if needed.
Add the meat mixture to the prepared casserole dish, then top with the mashed cauliflower and remaining cheese.
Bake for 30-35 minutes, or until heated through, and cheese is slightly browned.