Melt 1 tablespoon butter in a large skillet over medium-low heat, add the onions and 1/4 teaspoon salt and cook for 22 to 28 minutes, stirring often and adding water when the pan gets dry.
Transfer the browned onions to a bowl and raise the skillet heat to medium.
Mix the ground chicken with Worcestershire, remaining salt and black pepper, shape it into 12 thin slider-size patties and cook them in batches for 3 to 4 minutes per side, until each reaches 165°F.
Heat the oven to 375°F, slice the connected rolls horizontally and place the bottom slab in a baking dish.
Layer half the Gruyere, the 12 chicken patties, caramelized onions and remaining Gruyere over the rolls, then replace the tops and brush them with remaining melted butter.
Cover and bake for 8 minutes, uncover and bake for 4 minutes more, then rest for 5 minutes before cutting into 12 sliders.
