Bring a large pot of water to a boil. Add 1 tablespoon salt. Cook spinach spaghetti in the boiling water, stirring occasionally, until tender yet firm to the bite, about 12 minutes. Drain.
Place avocado, 2 tablespoons olive oil, coconut milk, and 2 tablespoons basil in a food processor. Process until smooth. Season pesto with salt and pepper.
Heat 1 tablespoon olive oil in a large skillet over medium heat. Add shallot and garlic; cook for 30 seconds. Stir in zucchini; cook until tender, about 5 minutes. Add spinach and cook until wilted, about 3 minutes. Add 1 tablespoon basil. Season vegetables with salt and pepper and mix until combined. Stir in the spinach spaghetti and pesto. Garnish with the remaining basil.