Spread 1 cut onion at the bottom of a slow cooker.
In a small bowl, whisk together 3/4 teaspoon salt, 1/2 teaspoon ground black pepper, 1/2 teaspoon paprika, and 1/4 teaspoon poultry seasoning.
Place 3 garlic cloves inside the chicken cavity.
Brush the chicken all over with 1 tablespoon melted butter and sprinkle the paprika mixture all over the chicken.
Place the chicken breast-side up into the slow cooker.
Pour 1/2 cup water into the slow cooker and sprinkle 1/2 teaspoon liquid smoke over the chicken.
Cook on High for 1 hour, then reduce heat to Low and cook for 8 hours.
Remove meat from bones, shred meat with two forks, cut onions in half, and return meat and onions to the slow cooker.
While the chicken is cooking, whisk together 3/4 cup mayonnaise, 2 tablespoons cider vinegar, 2 teaspoons white sugar, 1/2 teaspoon salt, 1/4 teaspoon ground black pepper, and 1/4 teaspoon cayenne pepper in a bowl until the sauce is smooth.
Cover the bowl with plastic wrap and refrigerate for at least 1 hour to blend flavors.
Pour 2 to 3 tablespoons of sauce over the chicken mixture.
Continue cooking on Low for 1 more hour.