Drain the tuna thoroughly, whether packed in olive oil or water. Transfer to a mixing bowl and use a fork to break the tuna into fine flakes. Leave some texture for contrast, if desired. Set aside.
Prepare the vegetables and herbs by finely dicing the celery and red onions, thinly slicing the green onions, and chopping the dill. Mince the cornichons. Add all these to the bowl with the tuna.
Incorporate the lemon juice, mayonnaise, Dijon mustard, and a dash of pickle juice. Season generously with salt and pepper. Mix thoroughly until evenly combined. Refrigerate the mixture for a minimum of 15 minutes to allow the flavors to meld.
When ready to serve, toast slices of bread if preferred. Spread a layer of mayonnaise on one slice and Dijon mustard on the other. Place a leaf of iceberg lettuce on the base slice, spoon over a portion of the chilled tuna salad, top with a slice of sharp yellow cheddar, then close with the remaining slice of bread. Cut in half and serve immediately.