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To make the pickled onion, heat the vinegar, spices and sugar in a small pan with 4 tbsp water until the sugar dissolves. Put the onion in a bowl. Strain the liquid onto the onion and leave to pickle for at least 20 mins or up to a day. The longer you leave it, the softer and more pickled the onion will become.
Scatter the ham evenly over one of the tortillas, then sprinkle the cheese over the top. Spoon some of the onions over the cheese and top with the remaining tortilla. The tortilla can be made up to a day ahead and left in the fridge ready to cook.
To serve, heat half the butter in a non-stick pan. Once foaming, carefully slide in the quesadilla. Cook on a low heat for about 2-3 mins, pressing with a spatula, until the base is nicely coloured. Turn the quesadilla over to cook the other side, adding the remaining butter to the pan. Lift the quesadilla on to a chopping board, cut into wedges, and serve scattered with the mint and a few pieces of onion, with the remaining onion on the side.