Sauté scallions in olive oil over medium-high heat.
Spread 1 tsp. mayonnaise on each slice of bread and top with a basil leaf.
Layer the cheese, turkey, scallions, avocado, and tomato on one slice of bread and cover with the other, mayonnaise side in.
Place on a panini press set to medium-high and heat until cheese melts, 3 or 4 minutes.