Lay 6 tortillas on the work surface, and divide the cheese, jalapenos, and chicken among the tortillas. (Its best to leave about an inch border on the edge of the tortilla uncovered to allow for the spread of the melting cheese.) Top with remaining tortillas and press gently to seal. Melt 1 tablespoon butter in a small skillet. Carefully place a quesadilla in the skillet and fry, turning once, until golden and little bubbles appear on both sides, 4 to 5 minutes in all. Repeat with the remaining quesadillas, wiping out the pan between batches if the butter burns.
While the quesadillas cook make the salsa: Halve the tomatoes and grate them on the largest holes of a box grater into a bowl, discarding the skins. Grate the onion into the tomato and stir in the cilantro, hot sauce and salt, to taste.
Cut quesadillas into 4 wedges with a pizza wheel or knife and serve with the salsa and avocado.