In a small food processor, add the drained tuna, mayo, lemon juice, green onions, dill, and a pinch of salt and pepper. Pulse for 5–10 seconds until light and spreadable, stopping while the mixture still has some texture. Don't over-pulse. Taste and adjust seasoning, then refrigerate while you prep the sandwiches.
Optional: Heat a skillet over medium-high heat. Lightly butter the outer sides of the bread. Toast each side until lightly golden (not browned), then set aside.
Spread about 1/2 tablespoon pesto on the inside of each slice of bread. On one slice, layer avocado. On the other, layer tomato. Evenly divide the tuna mixture across the sandwiches, then close them up. Slice each sandwich in half and serve immediately.
Notes
Storage: Sandwich is best enjoyed fresh after assembling. Store the tuna mixture in an airtight container in the fridge for up to 3 days. To keep avocado from browning, brush slices with lemon juice, wrap tightly, and use within 24 hours.