Preheat oven to 400F.
To make the pesto
Using a Bump and Grind mortar and pestle, smash the garlic into a coarse puree.
Add 1-1/2 bunches of basil and pound until it is coarsely chopped. Mix in the grated Parmesan cheese.
Add 1/3 cup of pine nuts and smash to break them up.
Slowly mix in 1/4 cup of olive oil. Season the pesto to taste with salt and pepper. Set it aside.
Heat 1 tablespoon of oil in a large ovenproof sauté pan over high heat.
Sprinkle the chicken breasts generously with salt and pepper. Cook the chicken breasts just until golden brown but not yet cooked through, about 2 minutes per side.
Remove the pan from the heat. Spread the pesto over the top of the chicken breasts.
Transfer the pan to the oven and bake until the chicken is just cooked through, about 8 minutes.
Meanwhile, to prepare the pasta: Cook the spaghetti in a large pot of boiling salted water over high heat until al dente, stirring occasionally to prevent it from sticking, about 8 minutes.
Drain the spaghetti, reserving 1/2 cup of the pasta cooking water in a large sauté pan over medium heat.
Whisk 1 tablespoon of olive oil into the reserved pasta cooking water to blend well. Add the spaghetti and parsley and toss to coat.
Using a two-pronged carving fork, swirl some of the pasta around the fork. Slide the pasta off the fork and mound it in the center of a plate. Repeat for each serving.
Cut each chicken breast crosswise and on a slight diagonal into 4 slices. Arrange the chicken slices alongside the spaghetti.
Garnish with the remaining ¼ cup of pine nuts, shaved Parmesan cheese, and torn basil. Drizzle the remaining 1 tablespoon of oil over and serve.