In a medium bowl, mix the Fabanaise, herbs, ketchup, relish and Worcestershire; season the sauce with salt and pepper.
Heat a cast-iron skillet, griddle or grill over medium-high. In another medium bowl, season the beef and form 8 large, very thin patties. Drizzle the skillet with oil; add 4 patties and cook, turning occasionally, 6-7 minutes. Top 2 of the patties with a slice of cheese during the last minute or so of cooking. Top cheeseburgers with another cooked patty to melt the cheese; transfer to a plate. Repeat with the remaining patties and cheese.