Tear the purslane into 2- or 3-inch pieces, stripping the little shoots from the main stems; discard the woodiest pieces.
Lay the tortillas on your work surface. Sprinkle some cheese on half of each tortilla, leaving 1/2-inch border. Put some purslane on top of the cheese, and drizzle about 2 teaspoons of the salsa verde on top. Top the purslane with the remaining cheese. Fold the tortillas in half.
Heat a medium nonstick skillet over medium heat. Add 2 teaspoons of the butter and melt until the foam just about subsides. Put in 2 of the tortillas and cook, pressing lightly with a spatula once or twice, until the underside is golden brown, about 2 minutes. Flip, and cook until the second side is brown and the cheese is melted, about 2 more minutes.
Transfer the tortillas to a cutting board. Wipe out the skillet. Repeat with the remaining butter and tortillas, lowering the heat a bit if the butter or tortilla browns too quickly.
Use a pizza wheel or sharp knife to cut the quesadillas into wedges and serve immediately.
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