Prepare the onions:
Slice the onion in half lengthwise and then thinly into half-moon shapes. Break apart the slices into individual pieces and finely dice 1 tablespoon to reserve for the dressing.
Heat olive oil in a large skillet over medium heat. Add the onions and sauté, stirring frequently, until they turn translucent and develop a golden hue, approximately 15 minutes. Adjust the heat to avoid excessive browning. During the final minute, stir in the apple cider vinegar to deglaze the pan. Remove from heat and set aside.
Create the Thousand Island dressing:
Combine vegan mayonnaise, ketchup, dill relish, pickle juice, garlic powder, paprika, salt, and pepper in a small mixing bowl. Add the reserved diced onion and whisk until smooth. Set aside.
Cook the burger patties:
Wipe the skillet clean after cooking the onions. Flatten the burger patties slightly to fit the size of the bread slices. For added flavor, coat one side of each patty with Dijon mustard before placing it mustard-side down in the skillet.
Cook the patties according to the package directions, reducing the time for the second side by one minute if the patties are thin. A hint of pink in the center is acceptable. Remove the patties and allow the skillet to cool briefly before wiping it clean.
Toast the bread and melt the cheese:
Arrange a slice of vegan cheese on each piece of rye bread. Heat the butter in the skillet over medium heat. Place the bread in the skillet, cheese-side up. Lower the heat to medium-low, cover, and allow the cheese to begin melting, approximately 5 minutes.
Assemble the sandwiches:
On two slices of bread, layer 1/4 cup caramelized onions, 1 tablespoon Thousand Island dressing, one burger patty, and sliced pickles if desired. Top with the remaining bread slices.
Press the sandwiches lightly with a spatula, increase the heat to medium, and cook for several minutes until the cheese fully melts and the bread achieves a golden, crispy texture.
Serve:
Transfer the sandwiches to a cutting board, slice in half, and serve warm with additional Thousand Island dressing on the side.
Store leftover dressing and caramelized onions in airtight containers in the refrigerator for up to five days.