In a large bowl, combine the chicken pieces with 1½ teaspoons salt, oregano, cilantro, black pepper, chopped bell pepper, minced red onion, chopped celery, mashed garlic, capers, halved olives, and the lime juice; mix well and let marinate for 30 minutes.
Heat a large heavy pot or Dutch oven over medium-high heat and add 3 tablespoons vegetable oil, reserving 2 tablespoons for later.
Add the brown sugar to the hot oil and cook until the sugar turns dark brown (carefully watch to avoid burning).
Carefully add the chicken to the pot (reserve the vegetables and herbs from the marinade while adding the chicken), stirring for about 1 minute to sear the pieces and coat with the caramelized sugar.
Add the reserved vegetables and herbs from the marinade, the diced auyama or carrot (if using), and the tomato sauce; stir to combine.
Add 2 cups of water, bring to a boil, then taste and adjust salt as needed.
Stir in the rice and stir frequently at first to prevent sticking. When the water has mostly evaporated, reduce heat to low, cover with a tight-fitting lid, and cook 15 minutes.
Uncover, gently stir to lift rice from the bottom, then cover and cook an additional 5 minutes. Check for doneness; if still firm, cover and cook another 5 minutes until rice is tender but not mushy.
Remove from heat and let rest briefly before serving.