Preheat oven to 350°F.
In a deep 12-inch cast iron skillet, Dutch oven, or other oven-proof pan with a lid, heat oil over medium-high heat. Season chicken on all sides with salt and pepper. When oil is hot, reduce heat to medium. Add chicken skin-side down, working in batches if necessary. Cook until browned, about 8-10 minutes.
Remove chicken and set aside.
Add onion and mushrooms to the pan; cook until softened, about 3 minutes. Stir in rice, garlic, parsley and rosemary; cook for 2 minutes, stirring constantly.
Add broth, lemon juice, ¾ teaspoon kosher salt and ¼ teaspoon black pepper. Use a wooden spoon to scrape up any of the browned bits from the bottom of the pan. Remove from heat.
Arrange chicken on top of the rice mixture. Cover with a tight-fitting lid.
Bake until rice is tender, 40-45 minutes. Remove chicken, fluff rice with a fork, and stir in frozen peas and basil. Cover and let rice rest for 10 minutes.
For crispier skin, place chicken under the broiler for 2-3 minutes while rice rests. Alternatively, shred the chicken with two forks, discarding the skin and bones.
Stir the meat back into the rice.
Garnish with additional fresh herbs, lemon zest, or Parmesan if desired.