Preheat a grill pan over medium-high heat. In a large bowl, combine the beef, allspice, cayenne, pepper flakes, garlic powder, onion powder, oregano, brown sugar, thyme, salt, and black pepper and gently mix until just combined.
Form the meat into eight 2-ounce patties, then press the center of each with your thumb to create an indent. coat the grill pan with cooking spray.
Grill the sliders indent side up until the internal temperature reads 150 degrees, 2 to 3 minuted per side. Place a slice of cheese on each patty once you flip them. The cheese will melt over the sides of the patties. Remove the patties to a plate and cover loosely with foil to rest.
Wile the patties rest, brush the buns with melted butter and grill until lightly toasted, about e minutes per side.
Assemble the sliders by putting a tablespoon-sized dollop of the ketchup on the top and bottom buns, topping the bottom bun with the patty, and finishing with one onion ring on top of each slider.
in a nonreactive medium bowl, combine the ketchup, lime juice, brown sugar, oregano, thyme, garlic powder, onion powder, all spice, cayenne, pepper flakes, and salt and mix until incorporated. Use immediately or transfer the ketchup to an airtight container and refrigerate for up to 4 days.