In a bowl, add a part of the drained tuna reserved oil, mayonnaise, lemon juice, chives, parsley, salt, and pepper. Gently stir to combine.
Lightly brush both sides of each slice of bread with oil. Grill until lightly charred, then remove from heat and let cool slightly.
With the grilled side facing up, spread ½-1 tablespoon of cream cheese over each slice. Divide tuna mixture on each slice of bread and spread. Top with 1-2 slices of cheese.
Return to the grill, close the lid and cook 2-3 minutes, until the bread is toasted and the cheese is melted. Put the rest of the tuna on top, sprinkle with black pepper and serve immediately.