Preheat oven to 400 degrees F (200 degrees C). Line a baking sheet with parchment paper.
Lay bacon strips flat in a single layer on the baking sheet.
Bake in the preheated oven until browned and crisp, about 20 minutes. Drain bacon slices on paper towels. Increase oven heat to 450 degrees F (230 degrees C).
Slice torta rolls in half; spread one side of each roll with 1 teaspoon mayonnaise and the other with 1 teaspoon pesto.
Arrange mozzarella sticks in a single layer on a baking sheet.
Bake mozzarella sticks in the hot oven until golden brown, 10 to 11 minutes.
Divide mozzarella sticks over the bottom halves of the rolls; top with lettuce, bacon, and tomatoes. Season with salt and pepper. Cover with the tops of the rolls. Brush oil over sandwiches; grill in a panini press until crisped and hot, 4 to 5 minutes.