For the pickled onions: Combine the onions, vinegar, sugar and salt to in a small saucepan and bring to a boil. Remove from the heat and let cool completely. Drain the pickled onions and set aside.
Meanwhile, prepare the beef: Combine the soy sauce, gochujang, sesame oil, sesame seeds, white vinegar, mirin and Asian pear in a blender and blend until smooth. Pour over the ground beef in a bowl and mix thoroughly.
Heat a large, heavy-bottomed skillet over medium-high heat. Add the beef and cook, breaking it up into pieces with a wooden spoon, until browned and cooked through, 8 to 10 minutes.
To serve: Heat a large cast-iron skillet over medium heat; add 2 tablespoons of the butter and let melt. Meanwhile, place 2 slices of the American cheese in a tortilla, then top with some of the ground beef, pickled onions and iceberg lettuce. Sprinkle with sesame seeds. Fold the tortilla in half to make a half-moon. Repeat with the remaining ingredients to make 5 more quesadillas. Working in batches, cook the quesadillas in a single layer, adding more butter as needed to the skillet, until golden and crispy on both sides, about 4 minutes per side.