Start by preparing the smoked mackerel. Check the fish carefully for any remaining bones and remove them. Flake the mackerel by hand into large pieces and set aside. Make sure the half red onion is sliced into paper-thin rings.
Whisk together a dressing of the extra virgin olive oil and the juice of the half lemon in a large bowl. Add the arugula or mixed greens, red onion rings, and capers to the dressing in the bowl. Toss everything together quickly and well so all leaves are evenly coated with the dressing.
Divide the prepared salad onto a large flat platter or directly onto plates. Distribute the reserved flaked smoked mackerel over the greens. Roughly tear the fresh dill and generously scatter it over the fish to balance the richness of the mackerel.
Serve the salad immediately after assembling. Bring the dish to the table right away so the greens retain their crisp structure.