In a small bowl, combine mayonnaise, basil, 1 tablespoon lemon juice, and 1 teaspoon garlic. Cover and refrigerate at least 2 hours.
In a large resealable plastic bag, combine olive oil, remaining 3 tablespoons lemon juice, remaining 3 teaspoons garlic, Italian seasoning, salt, and pepper. Add chicken; seal bag and toss gently to combine. Refrigerate for 1 hour.
Heat a medium skillet over medium-high heat. Remove chicken from bag, discarding marinade. Cook chicken until browned on both sides and cooked through, about 5 minutes per side.
Preheat panini press to medium-high heat. Spread mayonnaise mixture on cut sides of bread. On bottom half of bread, layer chicken, tomato, avocado, cheese, and top half of bread. Cut sandwich into fourths or sixths. Cook on panini press according to manufacturer’s instructions. Serve immediately.