Preheat a grill or grill pan to medium-high heat.
Brush the chicken breasts with olive oil and season both sides with salt and black pepper.
Grill the chicken for 5 to 7 minutes per side, or until fully cooked and the internal temperature reaches 165°F (74°C).
While the chicken cooks, combine avocado, basil, garlic, lemon juice, Parmesan, and nuts in a food processor or blender.
Blend until smooth, adding water a little at a time until the pesto reaches a creamy, spoonable consistency.
Let the chicken rest for 5 minutes, then top with avocado pesto and serve immediately.