Mince and mash the garlic to a paste, then transfer to a large bowl. Stir in the lime juice and oregano. Add the chicken pieces and rub all over with the marinade, until well coated. Cover with plastic wrap and leave at room temperature for 1 hour.
Spread the chicken over a sheet pan covered with paper towels and pat dry. Season with freshly ground pepper on both sides.
In a large heavy pan over medium heat, warm 1 tablespoon of the olive oil. Cook the chicken until lightly browned all over, about 3 minutes per side.
Using a slotted spoon transfer chicken to a bowl and cover with foil to keep moist. If there is too much fat left in the pan, drain a little. If there are too many pieces of garlic stuck to the pan, deglaze with about ½ cup of water, scraping the bottom of the pan and bring to boil. Pour and strain into a small bowl. Set aside.
Warm the remaining 1 tablespoon of olive oil in the same pot over medium heat and cook the sausage until lightly browned on both sides, about 2 minutes per side. Using the slotted spoon, transfer the sausage to the bowl with the chicken. Cover again.
Using the fat that’s left in the pan, reduce the heat to low and add the onions and scallions. Cook until soft and tender, mixing occasionally with a wooden spoon and scraping the bottom of the pan, about 2 minutes. Add the garlic and cook for another minute. Add the thyme, paprika, bay leaves, and tomatoes, and cook everything together until soft and tender, 4 to 5 minutes. Add the rice and stir well, making sure every grain is shiny and well mixed into the vegetable mixture. (If you would like to add the deglazing juices from the chicken, now is the time.) Then add the broth and bring to a boil. Reduce the heat to low, add the chicken, sausage, and any juice that accumulated in the bowl, arranging it all evenly over the rice. Season with salt and pepper, cover, and cook gently until the rice has absorbed all the liquid, 40 to 45 minutes.
Add the butter, mix gently, and garnish with the parsley just before serving.