Melt 2 tablespoons of butter in a large skillet over medium heat. Add the sliced onions and cook, stirring often, until they turn deep golden brown, about 20 minutes. Remove half the onions and set them aside for topping.
In a mixing bowl, combine the ground beef, breadcrumbs, egg, thyme, salt, and pepper. Mix until just blended, then shape into 8 small patties roughly the size of your slider buns.
Heat the remaining tablespoon of butter in the same skillet. Cook the patties for 4 to 5 minutes per side until browned and cooked through. Transfer them to a plate.
Sprinkle the flour over the onions left in the skillet. Stir for 1 minute, then slowly whisk in the beef broth. Simmer until the gravy thickens, about 3 minutes. Season with a pinch more salt and pepper if needed.
Place the cooked patties on the slider buns. Top each with reserved caramelized onions and a slice of Gruyere. Spoon gravy over the top and close the buns.