Prepare the salmon: Prepare a charcoal fire in an outdoor grill (alternatively, cook the salmon indoors on a grill pan); the coals should be medium-hot and the rack should be positioned about 6 inches from the fire. Place the salmon fillets on a cutting board, skin-side down. Cut down the center of each fillet lengthwise along the center seam. Cut through the flesh but not the skin. Draw the salmon flesh apart at the incision, flattening the fillets, leaving them skin-side down. Fold each fillet into a horseshoe shape, bending it back onto itself, so it resembles a boneless salmon steak. Brush each piece of fish lightly on both sides with extra-virgin olive oil and sprinkle with kosher salt and pepper.
Grill the salmon just until the flesh firms and grill marks appear, about 2 1/2 minutes on the first side and about 1 1/2 minutes on the second side. Rotate each fillet one quarter-turn (90 degrees) and grill just long enough to mark the fillets, completing a cross-hatch pattern, less than 1 minute. Turn the fillets over and repeat the grilling and rotating procedure until the other sides show the cross-hatch grill markings. Remove the fish from the grill and reserve. (The salmon can be grilled up to 1 hour in advance of the meal.)
Prepare the salad: Roll the cucumber pieces up in a clean linen napkin or towel, and then smash them, pressing down hard with the heels of your hands while ensuring the towel doesn't unroll, to release the juices. Transfer the crushed cucumbers to a cutting board, and then coarsely chop. Combine the cucumbers, sliced fennel, dates, walnuts, chives and some of the fennel fronds in a salad bowl and mix gently.
Finish cooking the salmon and dress the salad: Preheat the oven to 350 degrees F. Arrange the salmon fillets in a baking dish, brush each one lightly with extra-virgin olive oil and roast about 3 1/2 minutes for medium-rare.
Meanwhile, dress the cucumber salad with the lemon juice and 4 tablespoons of fruity olive oil. Then season with sea salt and pepper. Toss the salad lightly, and then arrange equal portions on each of six plates. Top each portion of salad with a grilled salmon fillet. Drizzle a little fruity olive oil on top of each portion, garnish each plate with fennel fronds and serve.