Take the mozzarella out of the refrigerator half an hour in advance and let it come to room temperature in its liquid. Cold buffalo mozzarella is tough and tasteless; at room temperature, it becomes creamy and releases its slightly tangy milk flavor.
Dry-toast the pistachios in a skillet until fragrant, let them cool, and roughly chop them.
Halve the galia melon, scoop out the seeds, and cut the flesh away from the rind. Cut it into thick wedges and arrange them across a large plate. Squeeze some lime juice over them; the acidity balances the sweetness of the melon and prevents the dish from tasting like a dessert.
Scatter the arugula among the melon and loosely drape the slices of Serrano ham over it. Don't lay them flat, but rather in folds; this keeps the ham airy and prevents it from sticking to the plate.
Tear the mozzarella into rustic chunks with your hands and scatter them over the plate. Drizzle everything with the olive oil and balsamic vinegar, sprinkle with the pistachios, and finish with plenty of torn basil, mint, and black pepper. You won't need salt, as the ham brings plenty of its own. Together, the ham and mozzarella provide the protein and fat, with the melon being the only noteworthy carbohydrate on the plate.