Melt the butter in a skillet over medium heat. Add the onions; cook until soft and golden, about 10 minutes. Raise the heat to medium-high and continue cooking, stirring, until uniformly brown, 5 to 10 more minutes. Season with salt and pepper and set aside.
Mix the pork, garlic and parsley in a bowl. Season the mixture with salt and pepper and form into 10 small patties, being careful not to pack them too tightly.
Heat a grill pan over medium heat; grill the patties until medium-rare, about 4 minutes per side. Place a cheese slice on the bottom half of each roll; top with a burger, some caramelized onions and the roll top.