In a food processor or blender, combine the kale, apple, garlic, ½ cup of cheese, 1 tablespoon of olive oil, salt, and pepper. Blend, scraping down the sides of the bowl if needed, until everything is combined and resembles a chunky pesto, 30 to 60 seconds.
Heat the remaining 1 teaspoon of olive oil in a medium skillet over medium heat. Spread roughly 2 tablespoons of the kale pesto onto one tortilla, sprinkle ¼ cup of cheese over the kale, and cover it with another tortilla. Place the quesadilla in the skillet and cook until golden brown, about 3 minutes. Flip it and cook for another 2 to 3 minutes. Repeat this step with the remaining tortillas, kale mixture, and cheese. You will have 2 quesadillas.
Let the quesadillas cool on a cutting board for 5 minutes. Cut each into 4 wedges and serve. If packing for lunch, let cool completely before placing in the bento box.