Prepare the tzatziki sauce: In a medium bowl, combine the plain Greek yogurt, grated and squeezed English cucumber, 1 clove of minced garlic, 1 tablespoon of chopped fresh dill, 1 tablespoon of olive oil, 1 tablespoon of lemon juice, and 0.5 teaspoon of kosher salt. Mix well, cover, and refrigerate until ready to serve.
Prepare the marinade: In a separate small bowl, whisk together the remaining 2 tablespoons of olive oil, the zest of 1 lemon, 2 tablespoons of lemon juice, 3 cloves of minced garlic, 2 tablespoons of chopped fresh dill, 2 tablespoons of chopped fresh parsley, 1 teaspoon of kosher salt, and 0.5 teaspoon of black pepper.
Marinate the fish: Place the salmon fillets in a shallow dish. Pour the marinade over the fillets, turning to coat evenly. Let marinate at room temperature for 15 to 20 minutes.
Preheat the grill to medium-high heat (around 400°F). Clean the grates thoroughly and brush lightly with oil to prevent sticking.
Place the salmon fillets skin-side down on the hot grill. Cook undisturbed for 4 to 5 minutes, allowing the skin to crisp.
Carefully flip the salmon and grill for an additional 3 to 4 minutes, or until the internal temperature reaches 135°F for medium doneness and the flesh flakes easily with a fork.
Remove the salmon from the grill and let it rest for 3 minutes.
Serve the grilled salmon warm, topped with a generous dollop of the chilled tzatziki sauce and extra lemon wedges on the side.