Mix the ground chicken with Worcestershire, 1/2 teaspoon salt and black pepper and shape it into 4 oval patties slightly larger than the bread.
Heat the oil in a skillet over medium-low heat, add the onions and remaining salt and cook for 20 to 25 minutes, stirring often and adding water when the browned bits threaten to burn.
Transfer the onions to a bowl, raise the heat to medium and cook the chicken patties for 5 to 6 minutes per side, until browned and 165°F in the center.
Layer 4 bread slices with 1 slice Swiss, 1 patty, one-fourth of the onions and another slice Swiss, then close with the remaining bread.
Butter the outsides and griddle the melts over medium-low heat for 3 to 4 minutes per side, pressing lightly, until the rye is crisp and both cheese layers melt.