Start by trimming any excess fat or silverskin from the filet mignon. Pat them dry with paper towels, then season generously with salt and pepper. This helps create a beautiful crust when searing.
Heat olive oil in a skillet over medium-high heat. Sear each steak for about 3–4 minutes per side until golden brown. Use an instant-read thermometer to check for medium-rare (130°F) or your preferred doneness.
In a separate skillet, heat butter and olive oil. Add scallion whites and mushrooms, cooking until mushrooms release their juices and start to brown. Stir in tarragon and scallion greens at the end for fresh flavor.
After removing steaks, add shallots to the same skillet and cook for 30 seconds. Pour in vermouth and simmer until reduced by half. Mix cornstarch with beef broth, add to pan, and cook until slightly thickened to about ½ cup.
Plate the steaks and spoon mushrooms on top. Drizzle the vermouth pan sauce over both for a rich, flavorful finish. This visually appealing presentation is perfect for a date night or special dinner.