Divide vegan ground meat into 4 equal pieces. Pick up one piece, transfer to a cutting board, and use your hands to press and shape it into a patty about ¼-inch wider than your bun. Repeat to form remaining patties. Put three fingers of one hand together, and press a shallow indentation into the center of each patty. Season with salt and pepper.
In a large skillet, heat oil over medium-high until shimmering. Add patties and cook, flipping frequently, until well browned and an instant-read thermometer inserted into the coolest part of the center of the patty registers 120 to 125 degrees for medium-rare, or 130 to 135 degrees for medium. Immediately add cheese slices to the top of the patties, and transfer them to a large plate.
Dress bottom buns as desired. Place a patty on top, and dress top bun as desired. Close cheeseburgers and serve.