Step 1Combine the onion, carrot, bay leaves, thyme, parsley stems, salt, pepper and approximately 1 quart of water in a medium sauté pan (about 9-inch). Bring to a simmer. Add the chicken and more water if needed so the chicken is just covered. Return to a simmer. Cover and cook over low heat until the meat is firm and no longer pink inside, and a thermometer inserted into the thickest part of the chicken reaches 160 degrees, 20 to 25 minutes.
Step 2Remove the pan from heat, cover (with the chicken still inside) and set aside for 10 minutes. Remove the chicken from the cooking liquid. Set the chicken aside to cool.
Step 3Remove the skin, bones and any visible fat. Dice the chicken to use for the chicken salad (you will need 2 cups diced chicken for the salad; any extra can be saved for another purpose). If not assembling the salad right away, cover and refrigerate the chicken until needed, up to 2 days.
Step 1In a large saucepan, combine the potaotes with enough water to cover by about 1 inch and a pinch of salt. Cover the pan and bring to a boil. Reduce the heat to a gentle simmer and cook until the potatoes are very tender (test by piercing the thickest potato with a fork), 25 to 30 minutes. Drain and set the potatoes aside until cool enough to handle.
Step 2While the potatoes are cooling, assemble the chicken salad. In a small bowl, combine the chopped onion with 1 tablespoon lime juice and set aside for 5 minutes (the lime juice will soften the intensity of the onion). In a medium bowl, toss together the chicken and mayonnaise. Stir in the red onion, along with 1 additional tablespoon lime juice, and mix well. Stir in the cilantro, and season with one-half teaspoon salt and one-eighth teaspoon pepper, or to taste.
Step 3Peel the potatoes while they are still fairly hot. Cut them in a few chunks and put them in a large bowl. Mash them thoroughly with a potato masher, not in a food processor, one or two potatoes at a time.
Step 4Add the olive oil to the potatoes and mix well. Add the remaining 2 tablespoons lime juice, along with 2 teaspoons pepper paste and one-half teaspoon salt. Taste, and add more pepper paste, lime juice or salt if desired. The dish tastes best when the potato mixture is slightly hot from the pepper paste, to balance the mild chicken salad.
Step 5Assemble the appetizers a short time before serving. The potato mixture is best at room temperature. The chicken salad can be served cold or at cool room temperature. The appetizers can be prepared in small or larger servings. For smaller servings, form thick, round potato patties using one-fourth cup mashed potato; for larger patties, use one-half cup.
Step 6Top the potato patties with chicken salad (2 tablespoons salad for smaller patties, one-fourth cup for larger). Garnish each portion as desired with quartered grape tomatoes, small pieces of hard-boiled egg, avocado slices, and parsley or cilantro leaves.