Preheat the oven to 425°F. Use a large oven-safe skillet.
Pat the 1 pound chicken dry. In a bowl or on a plate, toss the chicken with 2 teaspoons dried oregano, 1 teaspoon paprika, 1/2 teaspoon cayenne pepper, 1 tablespoon fresh thyme leaves, salt, and black pepper to taste.
Heat 2 tablespoons extra virgin olive oil in the oven-safe skillet over high heat until shimmering. Sear the chicken 3–5 minutes per side, until golden brown. Transfer the chicken to a plate and reserve any juices.
Reduce heat to medium. Add 3 tablespoons salted butter to the skillet. When melted, add 1/2 cup dry orzo, 1 cup dry rice, 2 thinly sliced shallots, 4 chopped garlic cloves, and the remaining 1 tablespoon fresh thyme. Cook, stirring frequently, until the rice and orzo are lightly toasted, about 3–4 minutes.
During the last 2 minutes of toasting, add the final 1 tablespoon salted butter and the slices from 1 lemon to the skillet. Allow the butter to brown slightly around the lemon slices, watching carefully to avoid burning.
Pour in 3 cups low-sodium chicken broth and stir to combine. Add 2 cups chopped spinach and season with salt and pepper. Bring the mixture to a boil, scraping up any browned bits from the pan.
Slide the seared chicken and any reserved juices back into the skillet, nestling the pieces into the rice mixture. Bring briefly to a simmer, then cover the skillet and reduce the stovetop heat to the lowest setting. Let cook for 10 minutes, until most of the liquid is absorbed but the rice is not fully done. If the mixture looks dry before this, add up to 1 more cup broth (total 3–4 cups) as needed.
Transfer the uncovered skillet to the preheated 425°F oven and bake 10–15 minutes, until the chicken is cooked through and the rice is tender (chicken should reach 165°F or no longer be pink in the center).
While the skillet is baking, combine 1/2 cup pitted green olives (crushed), 1/2 cup crumbled feta cheese, 1/2 cup fresh basil leaves, and 2 tablespoons lemon juice in a small bowl. Season with a little salt and black pepper to taste.
Remove the skillet from the oven. Spoon the olive–feta–basil mixture over the chicken and rice and serve immediately.