Heat oil in a nonstick skillet over medium heat. Add the garlic and anchovies, mashing with the back of a spatula, and cook for 1 minute. Add the breadcrumbs and cook, stirring until golden brown, about 3 minutes. Remove from heat and stir in the lemon zest and basil. Season to taste with salt and pepper. Set aside.
Make the spaghetti:
Bring a large pot of heavily salted water to a boil. Cook the spaghetti according to the package directions.
Whisk together the olive oil, Parmesan, and lemon juice in a large bowl.
Drain the spaghetti, reserving some of the cooking water. Add the spaghetti to the lemon sauce, tossing to coat. Add some of the cooking water, a little at a time, if the sauce is too dry. Season to taste with salt and pepper.