In a large bowl, combine the ground beef, ground pork, birria sauce, garlic powder, onion powder, smoked paprika, cumin, salt, and black pepper. Mix until well combined.
Divide the mixture into 4 equal portions and shape them into burger patties.
Preheat a grill or skillet over medium-high heat. Cook the patties for about 5-6 minutes on each side, or until they reach an internal temperature of 160°F.
During the last minute of cooking, place a slice of cheese on each patty and cover to melt.
Toast the burger buns on the grill or in a toaster until golden brown.
Assemble the burgers by placing each patty on a bun and topping with diced onions and chopped cilantro. Serve with lime wedges on the side.