re: Best way to cook a filet at home without a grill?

From America's Test kitchen (they have it up on Facebook today as a matter of fact):

A nicely charred thick-cut steak certainly looks appealing. But cutting into the steak to find that the rosy meat is confined to a measly spot in the center—with the rest a thick band of overcooked gray—is a great disappointment. We wanted to find a surefire method for pan-searing thick-cut steaks that could deliver both a flavorful crust and juicy, perfectly pink meat throughout. We found it was essential to sear the steaks quickly to keep the meat directly under the crust from turning gray. But we’d need to take an untraditional approach for these thick-cut steaks and sear them at the end of cooking, rather than at the beginning.

We began by moving the steaks straight from the fridge into a 275-degree oven, which not only warmed them to 95 degrees but also dried the meat thoroughly—dry meat is essential for a well-browned crust. At this temperature, when the steak met the hot skillet, our steaks developed a beautiful brown crust in less than four minutes, while the rest of the meat stayed pink, juicy, and tender. (less)

Serves 4
Rib-eye or filet mignon of similar thickness can be substituted for strip steaks. If using filet mignon, buying a 2-pound center-cut tenderloin roast and portioning it into four 8-ounce steaks yourself will produce more consistent results. If using filet mignon, increase the oven time by about 5 minutes. When cooking lean strip steaks (without an external fat cap) or filet mignon, add an extra tablespoon of oil to the pan. If desired, serve with a pan sauce, relish, or butter; see related recipes.

INGREDIENTS
2 boneless strip steaks (1 1/2 to 1 3/4 inches thick (about 1 pound each) (see note above)
Kosher salt and ground black pepper

1tablespoon vegetable oil

INSTRUCTIONS
1. Adjust oven rack to middle position and heat oven to 275 degrees. Pat steaks dry with paper towel. Cut each steak in half vertically to create four 8-ounce steaks. Season entire surface of steaks liberally with salt and pepper; gently press sides of steaks until uniform 1 1/2 inches thick. Place steaks on wire rack set in rimmed baking sheet; transfer baking sheet to oven. Cook until instant-read thermometer inserted in center of steak registers 90 to 95 degrees for rare to medium-rare, 20 to 25 minutes, or 100 to 105 degrees for medium, 25 to 30 minutes.

2. Heat oil in 12-inch heavy-bottomed skillet over high heat until smoking. Place steaks in skillet and sear steaks until well-browned and crusty, about 1 1/2 to 2 minutes, lifting once halfway through to redistribute fat underneath each steak. (Reduce heat if fond begins to burn.) Using tongs, turn steaks and cook until well browned on second side, 2 to 2 1/2 minutes. Transfer all steaks to wire cooling rack and reduce heat under pan to medium. Use tongs to stand 2 steaks on their sides. Holding steaks together, return to skillet and sear on all sides until browned, about 1 1/2 minutes. Repeat with remaining 2 steaks.

3. Transfer steaks to wire cooling rack and let rest, loosely tented with foil, for 10 minutes while preparing pan sauce. Arrange steaks on individual plates and spoon sauce over steaks; serve immediately.