For the bacon jam: Cook the bacon in a large nonstick skillet over medium-high heat, stirring occasionally, until browned and crisp and the fat has rendered out. Allow to cool, then chop the bacon.
In the same pan with the bacon drippings, add the onions and cook, stirring occasionally, until slightly softened. Add the brown sugar, vinegar and thyme; bring to a simmer. Reduce the heat to low and cook until the liquid is absorbed and the mixture has thickened to a jam-like consistency, about 20 minutes. Stir in the bacon; remove from the heat.
For the burger: Divide the beef into 4 equal portions and form into burger patties. Sprinkle both sides of each patty with salt and pepper.
Heat the oil in a pan or griddle over high heat. Cook the patties until golden brown and slightly charred on the first side, about 3 minutes. Flip over the patties and cook until golden brown and slightly charred on the second side or to the desired degree of doneness, about 4 minutes for medium-rare. Add 1 slice of American cheese and 1 slice of Cheddar to the top of each patty during the last minute of cooking, then top the pan with a basting cover or tent the patties with aluminum foil to melt the cheese.
Coat the bottom of the buns with mayo. Put the patties on the mayo-coated buns and top with the arugula and bacon jam. Sandwich the burger together with the tops of the buns and serve immediately.