Filet mignon is the most tender cut of beef, and you pay for the privilege of it gracing your plate. It's not a particularly flavorful cut, though, due to its lack of fat marbling. Of course, that translates to less saturated fat, cholesterol and calories, which is a good thing. Just season filet mignon well to make up for its mild flavor. Because the meat is so tender, it's perfectly suited to dry-heat cooking on the grill; a 1 1/2-inch thick steak is just thick enough to develop a nice crust to provide textural contrast with the succulent meat underneath.
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Pat coarse kosher salt all over the filet mignon at least 40 minutes before cooking it, but up to a few days in advance. Salt draws moisture out of the meat via a process called osmosis. Given enough time, the salty water is reabsorbed into the meat to effectively season it and help prevent dried-out beef. Without adequate time, the moisture stays on the surface and cooks off.
Scrape the grill thoroughly; burned-on residue imparts undesirable flavors and also contributes to food sticking to the grill. Grease the rack well with cooking oil on a clean cloth or with a wad of paper towels. Preheat the grill to medium-high; arrange for a cooler section, if you don't have one built in, like by piling coals to one side or attaching an upper grill rack.
Season the steak with additional herbs and spices. Freshly cracked black pepper is standard. Dried tarragon, thyme or rosemary is tasty on steak, or give your filet mignon a little kick with chili powder. Alternatively, make a beef rub with ingredients like paprika, dry mustard, garlic powder, onion powder, celery salt and chipotle powder.
Place the filet mignon over indirect heat and close the lid to cook it gently at first. Turn it every minute or so to encourage even cooking. Grill it to about 7 degrees below your final target internal temperature, which should be 120 degrees Fahrenheit for rare, 130 F for medium-rare, 140 F for medium, 150 F for medium-well or 160 F for well done. As a general guide, plan on a 1 and 1/2-inch-thick filet mignon needing about 10 minutes total, over indirect heat to come in around 7 degrees shy of medium. Use an instant-read thermometer to check your steak's progress.
Move the filet mignon over the hottest part of the grill to sear it, and to develop its crisp, well-browned crust. Because the steak is already mostly cooked, it should only need about 1 minute per side to crisp up nicely. Remove it from the grill once the outside looks good.
Let the meat rest on a plate for about 5 minutes before cutting into it. While it sits, a 1 1/2-inch-thick steak's internal temperature rises another 3 to 4 degrees, which, along with the final searing over high heat, is enough to bring the filet mignon to your target temperature without overcooking. The juices also settle during resting, which prevents them from seeping out while you're trying to enjoy a moist steak.