Heat 3 tablespoons of the oil in a large skillet over medium-high heat. Add the fennel and cook, stirring frequently, until it begins to become translucent, 3 to 4 minutes. Add the chicken and season with salt and pepper. Add the shallots and continue to cook until the chicken is browned on all sides, about 3 minutes.
Add the rice and cook, stirring, until it is toasted, about 2 minutes.
Stir in the broth and bring the mixture to a boil. Reduce the heat so the broth simmers; cover the skillet and simmer until the rice is tender, 15 to 20 minutes.
Bring a medium saucepan of water to a roaring boil.
Meanwhile, fill a small bowl with ice water. Set aside.
Cook the pine nuts in a medium skillet over medium heat, shaking frequently, until lightly toasted, 3 to 5 minutes. Keep an eye on them, as they can burn quickly.
When the water is boiling, add a handful of salt (you want the water as salty as the ocean), then add the basil leaves. Use a slotted spoon, small strainer or spider to press down the leaves, making sure they’re fully submerged in the water. Count to five, remove the leaves with the slotted spoon and immediately plunge them into the ice water for 1 to 2 minutes.
Remove the basil from the ice water and squeeze out excess water.
Put the toasted pine nuts in a food processor. (If you don’t have a food processor, a blender is fine.) Pulse to chop the nuts. Add the garlic and basil and pulse. Once a creamy, green paste forms, with the motor running, slowly pour in the remaining 1/2 cup of olive oil, processing until the oil is fully incorporated. Transfer the paste to a medium bowl and fold in 1 cup of the Parmesan with a rubber spatula. Season with salt and pepper. Set aside.
Remove the skillet from the heat and stir in the pesto and butter, if using. Serve the chicken and rice topped with fresh basil and the remaining 1/2 cup Parmesan.
Not all garlic cloves are the same size. If you have a really large garlic clove, you can use just one instead of two smaller cloves.