Make the vinaigrette: Combine all vinaigrette ingredients in a glass jar with a tight-fitting lid. Cover and shake vigorously until emulsified. Set aside.
Prepare the skillet: Heat 2 tablespoons olive oil in a large skillet over medium-high heat until shimmering and hot.
Sear the swordfish: Generously season fillets on both sides with salt and black pepper. Add to the hot skillet and cook undisturbed until golden brown and the fish releases easily, 4-5 minutes. Flip and cook another 4-5 minutes until just cooked through. The fillets should turn opaque about 1/3 of the way up from the bottom as they cook. Remove from heat and let rest.
Assemble the salad: In a large bowl, combine carrot ribbons, arugula, basil, and parsley. Drizzle with 2-3 tablespoons of vinaigrette, season with salt and black pepper, and toss gently to coat.
Serve: Divide salad among four plates and top each with a swordfish fillet. Garnish with lemon slices and serve with remaining vinaigrette on the side.