Preheat the oven to 400ºF. Stack the tortillas on a plate and cover with a damp paper towel. Microwave for 30 seconds to soften the tortillas.
Slice the string cheese in half length-wise. Spread ½ Tbsp pesto over the lower half of each tortilla, staying away from the edges. Place one piece of mozzarella on top of each.
Fold the sides of the tortilla in on top of the cheese, then begin rolling up from the bottom until the cheese is completely rolled up in the tortilla.
Place the roll ups on a parchment-lined baking sheet, seam side down. Brush the roll ups with a light coating of cooking oil.
Bake the roll ups in the preheated oven for about 13 minutes, or until they're lightly golden brown on the edges and the cheese is melted.
While the roll ups are in the oven, prepare the creamy tomato dip. Stir together the mayonnaise, tomato paste, and minced garlic.
Let the roll ups cool just enough to handle, then serve with the creamy tomato dip.