In a small bowl, whisk together 3/4 cup water and the peanut butter. In a large skillet, heat 1/4 cup extra-virgin olive oil over medium-high heat. Add the chicken, patted dry, and cook until no longer pink, 3 to 4 minutes. Transfer to a plate. In the same skillet, cook the carrots and 1/2 cup chopped scallion whites over medium heat for 30 seconds. Stir in the peanut butter mixture. Return the chicken and its juices to the pan and cook, stirring, until heated through; season with salt and pepper. Top with 1/4 cup chopped scallion greens and serve with cooked brown rice.