Pistachio pesto: add basil, pistachios, garlic cloves, and salt to a food processor or blender. Pulse to combine.
When things are chopped well, start slowly adding your olive oil while the processor/blender is running on low. This starts to form an emulsion and gives you the consistency you would want.
When the pesto is smooth and has a thickened consistency, remove it from the processor/blender to a bowl add parmesan cheese, and stir.
Taste the fully combined pesto and add any additional salt and pepper that you think it might need. Store in the fridge for a few days.
Sandwich assembly: break the ball of burrata in half over a medium-sized bowl and put all the interior cheese and liquid into the bowl. The strands of cheese inside of a burrata are called stracciatella. Note: If you are using mozzarella instead of burrata, slice 2 or 3 slices per sandwich.
Toast both sides of the sandwich bun if desired. Spread pesto on the bottom roll.
Top the pesto with strings of stracciatella or mozzarella slices and then sprinkle a tablespoon or two of chopped pistachios on top.
Layer Mortadella slices on top of the cheese. Try to fold each slice over instead of just laying them on top of each other. This creates a bit of a thicker layer of meat for each bite. Sprinkle more chopped pistachios if you want at this point and then place the top of the bun to close the sandwich.