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1
Place chicken, onion, bell pepper, garlic, salt, peppercorns, and water (chicken should be submerged 2 inches; add more water if necessary). Squeeze lime juice into pot and add spent lime halves to pot.
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2
Bring to boil over medium heat, then immediately reduce heat to medium-low and simmer until chicken is cooked through (chicken's temperature should read 160u00b0FF), about 20 minutes.
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3
With tongs, transfer chicken to large bowl. When cool enough to handle, shred chicken into bite-sized pieces, removing and discarding bones. You should have 4 cups shredded chicken.
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4
Strain broth and discard solids. Reserve 4 cups and store the rest for different use.
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5
Melt 2 tablespoons butter in large heavy-bottomed saucepan over medium heat. Add oil, onion, garlic, and salt and cook, stirring, until softened, about 5 minutes. Stir in tomato paste and cook until darkened, about 1 minute.
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6
Stir in rice and carrots and cook, stirring, until rice is completely coated with onion mixture, about 1 minute. Add broth and bring to a boil over medium-high heat.
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7
Continue to boil until most of the liquid has evaporated and you can see small bubbles on the surface of the rice. Reduce heat to the lowest setting, cover, and cook for 20 minutes until rice is done. Remove from heat and fluff rice with chopsticks or fork.
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8
Melt 2 tablespoons butter in a large skillet with high sides. Add sausage and ham and cook over medium heat until lightly browned, about 3 minutes. Transfer to plate.
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9
Melt remaining 1 tablespoon butter in now empty skillet. Add onion and piquillo peppers and cook until softened, about 5 minutes. Stir in tomato paste and cook until darkened, about 1 minute. Add ketchup and chicken and cook, stirring, until chicken is completely coated in mixture.
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10
Stir in rice, then reserved sausage mixture. Stir in olives, capers, and frozen peas. Cook, stirring occasionally to allow flavors to meld, 10 to 15 minutes. Season to taste with salt.
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11
Serve with grated Parmesan cheese and buttered white toast.
