In a medium bowl, combine crab meat and green onion.
Heat grill or grill pan on stovetop to medium to medium-high heat.
Very lightly butter one side of each tortilla. Lay tortillas butter side down. Then spread crab mixture over half of each tortilla. Top crab mixture with shredded cheese. Fold bare tortilla half over the top of the crab and cheese. Lay quesadillas on hot grill grates for 2 to 3 minutes per side, or until each side is nicely charred and cheese is melted.
Serve with smoky salsa for dipping, plus lime wedges for squeezing over the top. You'll have enough smoky salsa for dipping your favorite tortilla chips, too.
