Preheat the oven to 350°F (175°C).
Heat olive oil in a large oven-safe skillet or casserole dish over medium heat.
Add diced onion, carrot, and celery to the skillet and sauté for 5 minutes until softened.
Add minced garlic and sliced mushrooms, cook for another 3 minutes until mushrooms are tender.
Add ground beef to the skillet and cook, breaking it up with a spoon, until browned and no longer pink, about 7 minutes.
Stir in diced tomatoes with their juice, gluten-free beef broth, tomato paste, dried thyme, dried oregano, salt, and black pepper.
Bring the mixture to a simmer and cook for 5 minutes to combine flavors.
Stir in frozen peas, then remove from heat.
If desired, sprinkle shredded cheddar cheese evenly over the top.
Place the skillet or casserole dish in the preheated oven and bake uncovered for 25 minutes.
Remove from oven and let cool for 5 minutes before serving.