Make the herb oil: Combine the oil, garlic, thyme, rosemary, basil, and pepper flakes in a narrow saucepan over medium heat. As the heat builds, there will be considerable bubbling as the water cooks out of the garlic and herbs. When this dies down and the basil and garlic turn brown, kill the heat and steep until cool. Strain the oil into a jar and toss the solids. Use immediately or cover tightly and freeze for later use.
Make the sauce: Heat 3 tablespoons of the herb oil in a wide saute pan over medium heat. Add the tomatoes and 1 teaspoon of the salt to the pan, and cook for 5 to 7 minutes, breaking up the tomatoes as they soften with a wooden spatula or an old-school potato masher.
Reduce the heat, stir in the anchovies, and cook until the anchovies completely vanish from view, about 5 minutes more.
Meanwhile, place the spaghetti in a pan wide enough that the noodles can be submerged by just a couple inches of cold water seasoned with the remaining tablespoon of salt. Place over high heat and when the boil is reached, reduce heat to maintain a simmer and cook, stirring often, until al dente, 5 to 7 minutes.
Fish the spaghetti out of the water with tongs or a hand strainer, allowing most, but not all, of the water to drain away. Stir into the sauce and cook on low for 5 minutes. Finish with butter, Parmesan, and fresh basil. If sauce seems too dry, add a quarter cup of the pasta water and serve.