Cut avocados into large chunks and place in a medium bowl. Add lime juice, salt and pepper and mash to desired consistency. Stir in tomato, cilantro, jalapeño and garlic. Set aside.
Heat grill to medium-high heat. Melt butter in a cast-iron skillet set on grill rack. Add onion and salt and cook 10 minutes, stirring frequently. Carefully remove pan from grill.
Place 2 tortillas on grill and spread half of each with some of the onion. Grill 30 seconds. Sprinkle 1/2 cup of the cheese over each tortilla and fold to enclose filling. Grill 1 minute, then flip over and grill 30 more seconds. Repeat with remaining tortillas, onion and cheese. Cut each quesadilla into wedges and serve with guacamole.