Melt the unsalted butter in a large skillet or pan over medium heat.
Add the thinly sliced white onions and stir thoroughly so the pieces are evenly coated with the melted butter.
Cook the onions for approximately 1 hour and 15 minutes to 1 hour and 30 minutes, stirring periodically, until they become very tender and develop a rich golden caramelized color.
Reduce the heat if the onions begin browning more rapidly than desired. If the pan becomes dry during the long cooking process, stir in approximately 1 tablespoon of water as needed.
Near the end of the cooking time, pour in the apple cider vinegar. Continue stirring and cooking until most of the vinegar has evaporated.
Transfer the finished caramelized onions to a bowl and reserve them for assembly.
While the onions are cooking, place the mayonnaise, Dijon mustard, chopped parsley, and fresh lemon juice in a small bowl.
Stir until all ingredients are thoroughly incorporated and the sauce has a smooth, even consistency.
Refrigerate the prepared sauce until it is needed.
After removing the caramelized onions, use the same pan to prepare the beef hot dogs.
Set the pan over medium heat and add the hot dogs. Cook while turning them occasionally until the exterior develops crisp areas and is evenly browned in spots.
Arrange each cooked beef hot dog inside a hot dog bun.
Spoon some of the chilled parsley mayo sauce over each hot dog, then generously distribute the caramelized onions over the top.
Finish each hot dog with an even layer of shredded Gruyère cheese.
Transfer the assembled hot dogs to a baking sheet and place them under the oven broiler for a few minutes, cooking only until the Gruyère is fully melted and bubbling.
Remove from the oven once the cheese is melted, or allow it to broil slightly longer for a lightly golden surface.