For the Parsley Pesto, preheat the oven to 350 degrees Fahrenheit. Spread the walnuts out on a baking sheet and place it in the oven, toasting the walnuts until aromatic, about 8 to 10 minutes. Remove from the oven and allow to cool for 10 minutes before placing them in the bowl of a food processor or blender along with the remaining pesto ingredients *except* the olive oil. If you're using a blender, remove the capped center of the blender lid to leave a small hole in the cap. Blend at low speed while adding the olive oil in a slow continuous drizzle. Blend until fairly smooth, about 1 to 2 minutes at medium low speed, stopping to scrape down the sides of the blender as-needed. Transfer to a large heat-safe bowl, press plastic wrap onto the top of the pesto so it touches, and set it aside.
For the spaghetti with asparagus and caramelized spring onions, heat 4 teaspoons of the olive oil in a large sauté pan over medium heat. Add the spring onions, stir to coat in the oil, and reduce the heat to low. Continue cooking until the whites of the onions have turned very light gold, about 20 minutes, stirring every 3 to 5 minutes.
While they're cooking, you can prepare the toasted garlic breadcrumbs. Heat the olive oil in a medium sauté pan over medium low heat. Add the garlic and stir to coat in the oil. Continue cooking until softened, aromatic, and light gold around the edges. Add the breadcrumbs and salt and stir to combine. Spread out the breadcrumb mixture with the end of your spoon so that it covers the bottom of the pan evenly. Continue to cook until the breadcrumbs are toasted and deeply golden, about 1 to 2 minutes, stirring every 30 seconds. Remove from heat, transfer the breadcrumb mixture to a wide shallow bowl, and allow to cool. Once cooled, toss in the chopped fresh parsley.
When the spring onions are done caramelizing, remove them from the pan with a spatula and set them aside in a medium bowl. Cut the cut last 1/3 off the bottoms of the asparagus spears and compost it. Cut the remainder into roughly 1-inch long pieces. Add the asparagus, salt, and remaining 4 teaspoons of olive oil to the same pan you cooked the spring onions in and stir to coat them in the oil. Raise the heat to medium and cook until the asparagus' skin is slightly wrinkled and lightly golden in some places, about 8 to 10 minutes, stirring every minute or two. Remove from heat and transfer the asparagus to the bowl with the caramelized spring onions.
Bring a large pot of water to a boil with a few generous pinches of salt. Add the spaghetti and cook according to the package directions. Instead of straining the pasta, remove the plastic wrap from the top of the pesto. Use spaghetti tongs to grasp the pasta and transfer it to the bowl with the pesto. Add a 1 to 3 tablespoons of the pasta cooking liquid to the mixture, as needed, and mix until the pasta is evenly coated in the sauce. Add the asparagus, caramelized spring onions, and snap peas and toss to combine. Serve alongside the breadcrumbs to be sprinkled atop each serving.