Slice chicken breasts in half horizontally. Place the chicken in a Ziploc bag. Pour in the pesto and oil. Seal the bag and shake it up until the marinade is combined and the chicken is well coated.Preheat an outdoor grill or a skillet over medium-high heat. Grill the chicken for 6-7 minutes on each side until cooked through.
In a small bowl, whisk the mayo and pesto to form a creamy sauce.
To assemble: Spread 1-2 tablespoons of the pesto mayo on two slices of bread. Top with the sliced avocado, grilled chicken, a slice of tomato, and a slice of mozzarella.
optional: To toast, Heat 1 tablespoon butter in a pan or skillet over medium heat. Place sandwich on melted butter and heat until toasted golden and cheese is melted.