This mild, crunchy Brazilian chicken salad is a nice addition to a buffet; it could also be served on its own for lunch or a light supper.
In a small stockpot, cook chopped onion, garlic, and half the parsley in oil until slightly softened, about 5 minutes. Rub chicken with paprika, salt, and pepper and add to pot. Brown chicken on all sides. Add water to cover and bring to a boil, then lower heat to a simmer. Cook until meat is almost falling from bones, about 1 hour. Remove chicken from cooking liquid and set aside to cool. Strain cooking liquid and moisten chicken with a little of it. Discard remainder, or set aside for later use. (Chicken can be prepared a day ahead of time.)
In a large, deep, heavy frying pan, heat at least 3 inches of vegetable oil to 375 degrees. Remove 1 potato from water and grate lengthwise on the largest holes of a hand grater, pressing hard to get longest shreds possible. Sprinkle carefully and evenly into the hot oil, covering entire surface of oil. Do not stir. When brown around edges, turn as you would a pancake. Cook until deep golden brown, lifting up edge of pancake to check color. Remove from oil and drain on paper towels. Sprinkle with salt and pepper. When cool, crumble pancake with hands and set aside. Repeat with remaining potatoes, grating and frying 1 potato at a time.
When chicken has cooled, remove meat from bones and discard skin and bones. Shred chicken into long, thin pieces.
In a large mixing bowl, combine carrots, sliced onions, green beans, shredded chicken, fried potatoes, remaining parsley, and mayonnaise. Toss well to combine, season to taste with salt and pepper, and serve within 1 hour.