Slice the chicken breasts horizontally to make thinner cutlets.
Place the chicken in a zip-top bag or bowl and add 1/4 cup pesto and 1 tablespoon oil. Seal and shake or mix until the chicken is evenly coated.
Preheat an outdoor grill or a skillet over medium-high heat.
Grill or cook the chicken 6–7 minutes per side, until cooked through and an instant-read thermometer registers 165°F (74°C). Remove and let rest for a few minutes, then slice if desired.
In a small bowl, whisk together 1/4 cup mayo (or sour cream) and the remaining 1/4 cup pesto to make a creamy pesto mayo.
To assemble each sandwich, spread 1–2 tablespoons of the pesto mayo on two slices of bread. Top with sliced avocado, a cooked chicken breast, a slice of tomato, and a slice of fresh mozzarella.
Optional: For a toasted sandwich, heat 1 tablespoon butter in a skillet over medium heat and cook the assembled sandwich, pressing slightly, until golden and the cheese melts, about 2–3 minutes per side.