Preheat the oven to 350°F.
In a medium bowl, combine the tuna, avocado, carrots, dill, mustard, mayonnaise, salt, and pepper. Mix well.
Brush the olive oil onto the sourdough bread slices.
Place the bread slices on a baking sheet and bake in the preheated oven for 5 minutes to slightly toast.
Remove the bread from the oven and spread the tuna mixture evenly over each slice.
Top each slice with sliced tomato and mozzarella cheese.
Return the baking sheet to the oven and bake for an additional 5-7 minutes, until the cheese is melted and bubbly.
Serve immediately and enjoy!