In a bowl, whisk the vinegar, sugar, 2/3 cup water and 1 tsp. salt until dissolved. Pack the sliced onion into a 1-pint jar; top with the vinegar mixture. Let sit at room temperature about 1 hour.
Measure out 1/2 cup of the drained pickled onions; chop. (Refrigerate remaining onions up to two weeks.) Transfer to a bowl, then whisk with the mustard and honey; season relish with salt and pepper.
Heat a griddle pan, a large, heavy skillet or a grill over medium-high. Arrange 4 slices of cheese on a work surface. Pile one-quarter of the pastrami on top of each slice of cheese. Top the pastrami piles with the remaining slices of cheese.
Form the ground chicken into 4 patties (thinner in the centers for even cooking); season. Cook until cooked through, about 6 minutes per side. After flipping, top with pastrami and cheese; tent with foil to melt the cheese.
Spread some relish on the roll bottoms, then add the patties and roll tops. Serve with the remaining relish.