For the filling:
Trim the mushroom stems close to the caps. Put the mushrooms, shallots and garlic in a food processor and pulse until finely chopped but not puréed; scrape the bowl as needed. Transfer the chopped mushrooms to the center of a clean dishtowel. Gather up the sides of the towel and twist, keeping the mushrooms well contained in the cloth, bonbon style. With one hand, hold the ball of mushrooms over the sink; with the other hand, twist the gathered cloth, squeezing out as much liquid as possible.
Heat the butter and oil in a 10-inch skillet or sauté pan over medium heat. Add the mushrooms, cover, and cook, stirring occasionally, until the mushrooms are very soft and fragrant, 3 to 5 minutes. Don't let them brown. Uncover and cook until the pan is mostly dry, 2 to 3 minutes. Add the parsley; season with 1/2 teaspoon salt and pepper to taste. The filling can be refrigerated for up to a week or frozen for up to two weeks. If frozen, thaw overnight in the refrigerator and, if necessary, cook gently in an uncovered skillet to evaporate any juices that may have developed in the freezer.
For the croustades:
Sear the filets: Season the filets mignons generously on all sides with salt. Heat the butter and oil in a 10-inch skillet or sauté pan over medium-high heat until very hot and sizzling.
Put three of the filets in the pan and sear on one side until well browned, 1 to 2 minutes. Turn and brown the other side. Then quickly sear along the sides, using tongs to turn, about another 1 minute per side. Transfer to a plate lined with paper towels. Repeat with the remaining filets. Cover and refrigerate for at least 1 hour.
To assemble: Mash the Boursin with a fork in a small bowl until spreadable. Remove at least 24 sheets of phyllo from the package and cut them into
10-inch squares. Cover them with -plastic wrap and a damp dishtowel while you work to keep them from drying out. Lay a single phyllo sheet on a clean, dry surface. With a pastry brush, lightly paint an even coat of the melted butter over the entire surface of the square. Sprinkle with about 1/2 tsp. of the chives. Set a second sheet at a 90-degree angle over the first. Brush butter over it as well and sprinkle with another 1/2 tsp. chives.
Make four layers, omitting the chives from the last layer, with the corners of the phyllo pointing in different directions, starlike. Blot one filet dry with a paper towel, set it in the center of the star, and sprinkle it with salt. Spread about 2 Tbs. of the Boursin on the filet and top that with 2 generous Tbs. of the mushroom filling.
Pick a starting point and work your way around the filet, gathering the edges of the phyllo star together, beggar’s purse style. Lightly pinch together the gathered phyllo close to the surface of the meat to hold it in place. Pull the corners open slightly as if making a paper flower. Brush the whole exposed surface lightly with more butter, being sure that the bottom is buttered as well. Transfer to a heavy baking sheet. Repeat this “packaging’” for the remaining filets. If you’re working ahead, chill the filets uncovered on the baking sheet until ready to bake, up to 8 hours.
To bake: Take the croustades out of the refrigerator about 20 minutes before you’re ready to bake them. Position an oven rack in the lower third of the oven and heat the oven to 400°F. Bake the croustades, rotating the baking sheet after 10 minutes to ensure evening browning, until an instant-read thermometer inserted through the side of each packet and into the center of the filet reads 130°F for medium rare, 17 to 22 min. Serve immediately.