Process olives, basil, pine nuts and cheese in a food processor. Slowly add oil at the end of processing.
Rub Halloumi slices with olive oil and grill over medium heat for 1 minute per side. For presentation purposes cut the watermelon in round slices, roughly the size of the tomato slices using a cookie cutter or glass as a guide.
Using 4 chilled plates, stack 1 slice of tomato, then 1 slice of watermelon, 1 slice of grilled Halloumi, then top with another slice of tomato.
Finish with a dollop of Kalamata pesto.
Garnish each plate with arugula leaves, grated Parmigiano-Reggiano and olive oil.
Drizzle plates with a touch of balsamic reduction and top with basil leaves, whole or shredded.