Preheat the oven to 375°F (190°C).
Season the chicken thighs with salt and black pepper on both sides.
Heat olive oil in a large oven-safe skillet or Dutch oven over medium-high heat.
Add the chicken thighs skin-side down and cook for 5-7 minutes until the skin is golden brown and crispy. Flip and cook the other side for 3 minutes. Remove the chicken from the skillet and set aside.
In the same skillet, reduce heat to medium and add diced onion, minced garlic, diced carrots, and sliced mushrooms. Cook, stirring occasionally, for 5-6 minutes until vegetables are softened.
Sprinkle the flour over the vegetables and stir well to coat. Cook for 1-2 minutes to remove the raw flour taste.
Gradually whisk in the chicken broth and milk, stirring constantly to avoid lumps.
Add dried thyme, dried rosemary, paprika, and ground nutmeg. Stir and bring the sauce to a gentle simmer. Cook for 5 minutes until the sauce thickens.
Stir in the frozen peas and remove the skillet from heat.
Return the chicken thighs to the skillet, placing them skin-side up on top of the sauce and vegetables.
Sprinkle shredded cheddar cheese evenly over the casserole, then top with grated Parmesan cheese.
Cover the skillet tightly with aluminum foil if you plan to make ahead and refrigerate. Otherwise, bake uncovered.
To make ahead: Refrigerate the covered casserole for up to 24 hours. When ready to bake, preheat oven to 375°F (190°C) and bake covered for 30 minutes, then remove foil and bake an additional 15 minutes until cheese is golden and chicken is cooked through (internal temperature 165°F / 74°C).
If baking immediately, bake uncovered for 40-45 minutes until chicken is cooked through and cheese is bubbly and golden.
Remove from oven and let rest for 5 minutes before serving.
Garnish with chopped fresh parsley and serve warm.