Put all the chicken scraps and bones in a stockpot, along with the rest of the stock ingredients. Cover with water and bring to a boil. Simmer for 2 hours, adding water as necessary to keep the ingredients covered. Strain the stock discarding the solids. Cover and refrigerate.
Bring the broth to a simmer in a saucepan on top of the stove.
Heat 2 tablespoons of the butter and oil in a 4-quart pot over medium heat. Add the onion and sauté until it begins to soften. Add the rice and stir until all the grains are well coated.
Begin to add the simmering broth, 1/2 cup at a time, stirring frequently. Wait until each addition is almost completely absorbed before adding the next 1/2 cup, reserving about 1/4 cup to add at the end. Stir frequently to prevent sticking. At about the 12 minute mark, stir in the chicken so that will have time to heat through.
After approximately 18 minutes, when the rice is tender but still firm, add the reserved broth and the prepared pesto. Turn off the fire and add the remaining tablespoon of butter. Stir vigorously to combine with the rice. Serve, garnishing each serving with pine nuts.