Combine the lemon zest, garlic, 1 cup olive oil, the thyme and Ottoman spice in a bowl; mix well. Divide the marinade between two 1-gallon zip-top bags, then divide the swordfish between the bags. Let marinate in the refrigerator for at least 1 hour or up to overnight.
Preheat a grill to high and oil the grates. Grill the red onion and jalapeno until marked, about 2 minutes per side. Transfer the red onion to a bowl, top with the vinegar and cover with plastic wrap, trapping the steam inside to finish the cooking process. Peel, seed and chop the jalapeno.
Make the salsa verde: Combine the jalapeno, parsley, cilantro, mint, lime juice and remaining 1 cup olive oil in a bowl. Once cool enough to touch, remove the onion from the vinegar and coarsely chop, then add both the vinegar and the onion to the salsa verde; season with 1/2 teaspoon kosher salt and combine well.
Remove the swordfish from the marinade, shake off the excess and season with kosher salt. Reduce the grill heat to medium. Grill the swordfish 2 minutes, then rotate 90 degrees and grill 2 more minutes to create crosshatch grill marks, then flip and grill until just opaque, about 2 more minutes. Increase the grill heat to high. Grill the halved lemons and the avocados cut-side down until marked, 1 to 2 minutes.
Dress the watercress with some of the salsa verde on a platter. Top with the swordfish, avocado and more salsa. Garnish with the grilled lemons. Finish with a sprinkle of sea salt.