Preheat the oven to 375ºF. Heat the oil over medium heat in a deep oven-safe skillet.
Pat the chicken dry with paper towels, then season on all sides with the salt, pepper, and smoked paprika.
Add the chicken to the skillet and cook for about five minutes on each side, until crisp and just about fully cooked. Remove from the skillet and set aside.
In the same skillet, add the shallot and cook, stirring often, until softened. Add a splash of broth to scrape up the fond. Add the garlic, dried thyme, and dried rosemary, and cook, stirring often, until fragrant.
Add the cubed butternut squash, mix to combine, and cook, stirring occasionally, until slightly softened.
Add the rice, stirring continuously until lightly toasted. Season with a few pinches of salt. Pour in the broth, coconut milk, and lemon juice. Mix to incorporate. If the rice is not covered in liquid. Make sure the rice is submerged (if not, add water).
Place the chicken on top of the rice, top with the fresh rosemary sprigs, and bake until the rice has absorbed almost all of the liquid and the chicken reaches an internal temperature of 165ºF, 23 to 27 minutes.