Take the coppa out of the refrigerator fifteen minutes before serving and lay the slices separately on a plate. Cured ham has finely marbled fat that only softens and releases its nutty aroma at room temperature; straight from the fridge, the fat is hard and the ham tastes flat and salty.
Toast the hazelnuts in a dry skillet until the skins crack. Rub them in a clean kitchen towel to remove the bitter skins, let them cool, and chop them coarsely.
Halve the figs. Heat a grill pan until very hot and place the figs cut-side down in the dry pan. Leave them to sear for two minutes without moving them. Their natural sugars will caramelize against the hot metal into a dark layer, while the flesh underneath remains warm and juicy. Do not add oil, as it prevents the cut surface from making direct contact.
Scatter the arugula loosely over a large plate and drape the coppa slices over it in folds, keeping the ham light and airy so it doesn't stick flat to the plate.
Arrange the warm figs in between and drizzle everything with the olive oil and a few drops of balsamic vinegar.
Shave the pecorino over the top in wide ribbons using a vegetable peeler, sprinkle with the hazelnuts, and finish with thyme leaves and black pepper. No salt is needed, as the ham and pecorino bring plenty of it.