For the tenderloin: Preheat the oven to 400 degrees F.
Dry the tenderloin well with paper towels and sprinkle evenly with the salt and pepper. Heat a large ovenproof skillet over medium-high heat. Add the oil to the hot skillet followed by the tenderloin. Cook, turning every 4 minutes, until the tenderloin is an even, deep golden brown, about 12 minutes. Place the skillet in the oven until an instant-read thermometer reads 120 degrees F, or about 30 minutes for medium-rare . Remove to a platter to rest for 10 minutes before slicing.
For the sauce: Meanwhile, preheat a grill for cooking at medium-high heat.
Sprinkle the scallions with 1 teaspoon salt and 2 teaspoons olive oil. Place on the hot grill and cook until wilted and slightly charred, about 3 minutes per side. (Place a pan lid on top of the scallions if they are not browning enough.) Remove the charred scallions to a cutting board and roughly chop. Add the charred scallions to a bowl along with the mustard, vinegar, remaining 1/3 cup olive oil and remaining 1/2 teaspoon salt. Stir to combine.
Sprinkle the tenderloin with flake salt and serve with the scallion sauce.