Preheat the oven to broil. Line a baking sheet with parchment paper and set aside.
In a bowl, combine the tuna, 1 ½ tablespoons mayonnaise, mustard, carrots, dill, salt and pepper. Mix well to combine. Set aside.
Spread ½ teaspoon of remaining mayonnaise on one side of each slide of bread. Place the bread with mayo side down and assemble the tuna melts: top each slice of bread with mashed avocado, the tuna mixture, tomato slices, and cheese.
Add olive oil to a large pan over medium heat. Place the tuna melts on the pan and cook for about 5 minutes, until the bread gets brown and crispy.
Transfer the tuna melts to the lined baking sheet and broil for about 2-3 minutes, until the cheese has melted. Serve immediately.