Combine the olive oil, butter, sliced garlic, and white parts of the scallions in a small pot over low heat. Cook until the garlic and scallions are totally tender, almost melted into the combined fats, 10 -15 minutes. (If necessary, lower the heat even further to maintain a subtle simmer—you don't want the garlic or scallions browning.)

Add the anchovies and continue to cook on the lowest heat possible, stirring occasionally, until they're totally broken down into a nice golden paste, smelling a little nutty (not unlike browned butter), 15—20 minutes.

Add the green scallion tops and raw grated garlic. Continue to cook until the scallion tops are wilted and bright green and the garlic is fragrant, another minute or so. Remove from the heat, add the parsley, and season with salt and pepper as needed. Serve with vegetables of your choosing.