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 the chicken on all sides with salt and pepper to taste. When the oil is hot, reduce heat to medium. Add the chicken to the skillet skin-side down (working in batches, if necessary). Cook until the chicken is browned, about 8-10 minutes. Remove chicken from the skillet and set aside on a plate.
Add the onion and mushrooms; 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. Remove from heat.
Arrange chicken in rice mixture. Cover with a tight-fitting lid.
Bake, covered, until rice is tender, 40-45 minutes. Remove chicken, fluff rice with a fork, and stir in frozen peas and basil. Cover with the lid again and let the rice stand for 10 minutes while you handle the chicken.
If you like a crispier skin on the chicken, you can place the thighs under the broiler for a few minutes to help them crisp up again. Alternatively, you can shred the chicken, discard the skin and bones, and stir the meat back into the pot with the rice.
Garnish with chives, lemon zest, or Parmesan if desired.