Heat the olive oil and butter in a 12-inch cast iron (or other oven-safe skillet) over medium heat until the butter has melted. Add the anchovy paste and cook, stirring, until it has melted into the oil and butter, about 10 seconds. Add the panko crumbs and cook, stirring occasionally, until lightly golden brown, about 5 minutes. Transfer the crumbs from the pan to a mixing bowl and wipe the pan clean.
To the panko crumbs, add the parsley and lemon zest, and stir to combine.
Place the salmon filets on a board, skin-side down, and sprinkle lightly with salt and pepper. Brush the mustard on the top of the filets. Divide the crumb mixture evenly on top of the filets, pressing down.
Heat the vegetable oil in the skillet over medium-high heat. Add the salmon, skin-side down, and cook for 2 minutes. Transfer the pan to the oven and cook for 4-6 minutes or until the salmon is cooked through.