Preheat a grill to medium high. Put the eggs in a medium saucepan and cover with cold water. Bring to a boil, then remove from the heat and let sit, covered, 10 minutes. Drain and rinse under cold water until cool. Peel, chop and set aside.
Meanwhile, drizzle the salmon with the juice of 1/2 lemon and season generously with salt and pepper; set aside 10 minutes.
Make the dressing: Combine the juice of the remaining 1/2 lemon in a small bowl with the lemon zest, vinegar, shallot and tarragon; whisk to combine. Gradually whisk in the olive oil until combined; season with salt and pepper. Set aside.
Brush the grill grates with vegetable oil. Lightly brush the lettuce, avocados and corn with vegetable oil. Grill the vegetables, turning, until well marked, about 2 minutes for the lettuce and avocados and 5 to 6 minutes for the corn; transfer to a cutting board. Meanwhile, grill the salmon, turning once, until just cooked through, 6 to 8 minutes. Transfer to the cutting board and let rest 5 minutes.
Roughly chop the lettuce, discarding the core; transfer to a large platter. Add the basil and dill and toss. Cut the kernels off the corn and chop the avocado and salmon. Arrange the corn, tomatoes, hard-boiled eggs, salmon and avocados in rows over the salad; season with salt and pepper. Drizzle with the dressing and top with parsley.