Heat the oil in a large frying pan, add the shallot and cook for 3 minutes until softened. Add the mushrooms and cook for a further 3 minutes until golden. Add the garlic and cook for 1 minute, then season.
Arrange the mushroom filling over half of each tortilla, sprinkle over the Cheddar and fold over.
Heat a frying pan until smoking hot. Place 1 quesadilla in the pan and cook for 1–2 minutes until lightly crisp. Carefully turn the quesadilla over and cook for a further 1–2 minutes until the cheese has melted. Remove from the pan and keep warm. Repeat with the remaining quesadillas.
Meanwhile, make the salsa by mixing together all the ingredients in a bowl.
Cut the quesadillas into wedges, then spoon over a little salsa and serve sprinkled with the feta.