Start by making the onion topping: Melt the butter in a skillet over medium heat. Add the onions, garlic, and thyme and season with salt and pepper. Cook 30 minutes, stirring occasionally. Be careful not to brown the onions too fastyou want them to be very soft and caramel in color. Deglaze with the sherry, then stir in enough stock to keep the onions wet. Reduce the heat to low and keep warm.
Season the beef with salt and pepper and mix thoroughly. Score the mixture into 4 equal portions, then form each portion into 3 patties, making them thinner at the center than at the edges for even cooking and to ensure a flat surface (burgers plump as they cook). Drizzle the patties with EVOO.
Heat a cast-iron pan or griddle over medium-high heat. Cook the sliders, turning once, 5 to 6 minutes for medium-rare, a few minutes longer for medium-well. Top the sliders with cheese for the last minute or two of cooking, tenting the pan with aluminum foil, if you like, to help melt the cheese.
Place the sliders on the roll bottoms and top with onions, fries, and horseradish fry dipper. Set roll tops in place.