In a large pot, place the duck pieces and cover with water. Bring to a boil, then reduce the heat and simmer for 1.5 to 2 hours, or until the duck is tender and cooked through.
Remove the duck from the pot and let it cool. Once cool enough to handle, shred the meat, discarding the bones and skin. Set aside.
In a large skillet or Dutch oven, heat the olive oil over medium heat. Add the chopped onion, garlic, and diced carrot, and cook until the onion is translucent and the carrot is tender.
Stir in the sliced chorizo or Portuguese sausage and cook for another 2-3 minutes.
Pour in the white wine and cook for 2-3 minutes, allowing the alcohol to evaporate.
Add the rice to the skillet and stir to coat it with the oil and flavorings.
Pour in the chicken or duck broth, and add the bay leaf, dried thyme, salt, and black pepper. Bring to a boil.
Reduce the heat to low, cover the pot, and simmer for 15-20 minutes, or until the rice is cooked and the liquid is absorbed.
Stir in the shredded duck meat, frozen peas, and chopped parsley. Cook for an additional 5 minutes, until the peas are heated through.
If using cheese, sprinkle it on top and stir until melted and well combined.
For a crispy top, sprinkle breadcrumbs over the rice and place the skillet under the broiler for a few minutes until golden brown.
Enjoy your meal!