In a medium mixing bowl, combine the mayonnaise, lemon juice, dill and chives. Stir in the tuna, pickle relish and celery, using a fork to break up chunks of tuna.
Heat a large skillet on the stove over medium heat.
Spread a thin layer of mayonnaise on one side of each slice of bread. Place two slices, mayo side down, into the hot skillet.
Divide the tuna onto the two slices of bread and top with sliced tomatoes and cheese. Top with the other two slices of bread, mayo side up.
Grill for several minutes or until golden brown. Carefully flip with a large spatula and grill the other side until golden. Transfer to a cutting board and let rest for at least 5 minutes before serving.
Makes 2 sandwiches and can easily be double or tripled to make as many as you need.
Sandwiches are pretty filling and could also be divided to feed 4.