Steps:
- In a glass bowl, stir together the olive oil, lime juice, and marjoram. Add the chicken strips and toss to coat evenly. Cover and marinate in the refigerator for at least 4 hours or up to 24 hours.
- To prepare pesto, put all of the ingredients in a blender of in a food processor fitted with the metal blade and process until smooth. You may prepare this pesto up to 2 hours before serving, but do not expect it to keep overnight, as cilantro is fragile and its color changes and flavor fades within hours. Pesto ingredients are from cilantro thru pepper.
- Prepare the grill, place chicken in grill basket.
- Heat oven to 250
- Remove the chicken from the marinade and place on the grill rack about 6 inches above medium-hot heat. Grill quickly, turning once, until done, 1-3 minutes on each side. Remove to a dish and cover to keep warm.
- To cook the quesadillas, put the butter in a cast-iron skillet over medium-high heat. When the butter sizzles, put in 1 tortilla, let it crisp on one side, about 2 minutes, and then turn it over. Spoon 2 tablespoons of the pesto, 3 tablespoons of the cheese, and one fourth of the chicken onto one half of the tortilla, then fold over the other half to cover the filling. Continue cooking the quesadilla until crisp and the cheese melts, about 5 more minutes. Remove the quesadilla to the warm oven to hold while you good the others.
- Serve piping hot with salsa on the side, if desired.
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