Preheat the oven to 375°F (190°C).
Heat olive oil in a large skillet over medium heat.
Add diced onion and cook for 3-4 minutes until translucent.
Add minced garlic and cook for 1 minute until fragrant.
Add ground beef to the skillet, breaking it apart with a spatula. Cook until browned and no longer pink, about 7-8 minutes. Drain excess fat if necessary.
Stir in diced zucchini, chopped cauliflower, and sliced mushrooms. Cook for 5 minutes until vegetables start to soften.
Add dried oregano, dried thyme, smoked paprika, salt, and black pepper. Stir well to combine.
In a separate bowl, whisk together heavy cream, sour cream, and beef broth until smooth.
Pour the cream mixture into the skillet with the beef and vegetables. Stir to combine and cook for 2-3 minutes until slightly thickened.
Remove the skillet from heat and stir in 3/4 cup shredded cheddar cheese until melted and well combined.
Transfer the mixture into a greased 9×9-inch casserole dish.
Sprinkle the remaining 1/4 cup shredded cheddar cheese and grated Parmesan cheese evenly on top.
Bake in the preheated oven for 20-25 minutes until the casserole is bubbly and the cheese on top is golden brown.
Remove from oven and let cool for 5 minutes before serving.