Chop the broccoli into very small florets. You can also use the broccoli stem - just peel it and chop roughly. Grate the cheese on the big holes of a box grater. Finely chop the garlic.
Heat 2 Tbsp of oil on a big frying pan. Add the chopped broccoli. Cook over high heat, stirring from time to time until the broccoli is tender, browned but still a little crunchy. It will take about 10 minutes. If your broccoli is already browned but still tough you can add a small amount of water to the pan to speed up the cooking process.
Add the chopped garlic and basil. Cook, stirring, for about a minute. Season to taste with salt and pepper.
Transfer the broccoli to a plate.
Assemble the quesadilla: If you have small tortillas: Sprinkle 1/3 cup of loosely packed shredded cheese over the tortilla, top with 1/2 cup of cooked broccoli and sprinkle another 1/3 cup of shredded cheese (spread all the ingredients in an even layer). Place another tortilla on top. If you have bigger tortillas: Top 1/2 of one big tortilla with the previously mentioned ingredients. Fold the tortilla in half. These quesadillas are easier to flip over, it is very difficult to flip over big double-tortilla quesadillas.
Repeat with the remaining ingredients.
Cook the quesadillas: Clean the pan with a paper towel and heat 1 Tbsp of oil. Place the quesadilla in the pan. Move it around to make sure it doesn't stick. Cook for a 1-2 minutes, or until golden brown then flip over using two spatulas and cook on the other side until browned and crispy (you can tell that the quesadilla is ready to flip over when the edges are golden, insert a spatula under the tortilla to see if it's browned).
Transfer the quesadilla to a chopping board. Sprinkle with a little salt. Cut into 8 parts / triangles (a pizza cutter is great for that).
Serve with tomato salsa and enjoy!