To prepare steaks: Dry-brine the steaks by rubbing the salt evenly over the surface. Transfer to a ziplock bag and then chill for at least 1 hour or up to 24 hours.
To prepare the salsa verde: In a food processor, combine the garlic, anchovies, and capers and pulse until finely chopped. Scrape down the sides of the processor bowl; add the vinegar, mustard, and parsley; and whiz while slowly drizzling in the 1⁄3 cup olive oil until smooth and bright green. Transfer the mixture to a lidded 8-ounce jar. Add the remaining 1 tablespoon olive oil over the top to create a film that will preserve the lovely bright green. Seal the jar tightly. Sauce can be made and chilled for up to 1 week. Remove the salsa verde from the cooler and let stand at ambient temperature for up to 2 hours before serving.
To Cook Steak: Heat the grill to high heat and grill steaks over the hottest part of the fire, flipping the steak every few minutes for even cooking; the cooking time will vary depending on thickness, about 10 minutes. Look for an evenly caramelized and browned outer crust to check doneness. Slight char is okay but don’t let the crust burn! Use a thermometer to check that the internal temperature of the thickest part of the steak is between 125° and 130°F for medium-rare to medium (see chart in article for alternative temperature).
Transfer the steak to a cutting board and let rest for 10 minutes so the juices can redistribute. Slice the steak against the grain into 1⁄2-inch slices.
Serve directly from the cutting board with salsa verde for topping the steak.
If you steak is excessively fatty you can trim some of it to prevent flare-ups when the fat drips onto the coals. Skirt steak isn’t very fatty, but still, leave any fat on for flavor and just be prepared for flare-ups.
Store leftovers in an airtight container, chilled, for up to 3 days.