Preheat the oven to 350°F (175°C). Line a baking sheet with parchment paper or foil; if using foil, lightly spray it with cooking spray.
In a medium bowl, combine the softened cream cheese, basil pesto, sliced green onions, and shredded cooked chicken. Stir until evenly mixed and season with salt and pepper to taste.
Unroll the crescent roll dough. Press two triangle pieces together along the long edges to form a rectangle.
Spoon about 1/3 cup of the chicken mixture into the center of each rectangle. Bring the corners together and pinch to seal, forming a pillow. Repeat until filling is used.
Pour the melted butter into a shallow bowl and spread the panko crumbs on a plate. Dip or brush each pillow with melted butter, then roll it in the panko to coat evenly.
Place the coated chicken pillows seam-side down on the prepared baking sheet, leaving a little space between them.
Bake for 20–25 minutes, or until the pillows are golden brown and heated through. Remove from the oven and serve warm.