Preheat skillet over medium heat. Over the sink, squeeze thawed spinach until liquid no longer comes out. You can do this with your hands or place it in a kitchen towel and ring it out. It may stain the towel so use one you that can be stained.
In a large bowl, combine the spinach, cream cheese, chopped artichokes, parmesan, mozzarella, garlic, salt and pepper. Spread mixture evenly between 4 to 5 flour tortillas (depending on how full you want them). Top each with another tortilla. Set aside.
Melt 1 teaspoon of the butter in the skillet and add 1 prepared quesadilla. Cook until golden brown, 1 to 2 minutes, then flip and cook an additional 2 minutes. Remove from skillet and slice with pizza cutter. Cook remaining quesadillas, adding more butter when needed.